Minors and Cofactors: Formula, Application, Examples, Determinants

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As the name suggests, a Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs.", "about": [ { "@type": "Thing", "name": "Maths" }, { "@type": "Thing", "name": "SchoolLearning" }, { "@type": "Thing", "name": "Class12" }, { "@type": "Thing", "name": "Dsa" }, { "@type": "Thing", "name": "MathsClass12" }, { "@type": "Thing", "name": "MatrixAndDeterminant" } ] }</script> <script type="application/ld+json"> { "@context": "https://schema.org", "@type": "WebSite", "url": "https://www.geeksforgeeks.org/", "potentialAction": { "@type": "SearchAction", "target": "https://www.geeksforgeeks.org/search/{search_term_string}/", "query-input": "required name=search_term_string" } }</script> <script type="application/ld+json"> { "@context": "https://schema.org", "@type": "Organization", "name": "GeeksforGeeks", "url": "https://www.geeksforgeeks.org/", "logo": "https://media.geeksforgeeks.org/wp-content/cdn-uploads/20200817185016/gfg_complete_logo_2x-min.png", "description": "A computer science portal for geeks. 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Then dividing adjoint by determinant to get the inverse of the matrix." } }, { "@type": "Question", "name": "Can Minors and Cofactors be used to solve systems of Linear Equations?", "acceptedAnswer": { "@type": "Answer", "text": "Yes, minors and cofactors can be used to solve systems of linear equations by representing the system in the form of a matrix and using Cramer's rule, which expresses the solution in terms of determinants of smaller matrices obtained by replacing one column of the coefficient matrix with the constant terms of the system." } } ] }</script> <script> var arrPostCat = new Array(); var arrPostCatName="Mathematics"; arrPostCat.push('4685'); arrPostCat.push('6263'); arrPostCat.push('4667'); arrPostCat.push('4677'); var tIds = "4685,6263,4667,4677,5967,6312"; var termsNames = "Class12,SchoolLearning,DSA,Mathematics,MathsClass12,MatrixandDeterminant"; var tIdsInclusiveParents = "4685,6263,4667,4677,5967,6312" var domain = 1; var arrPost = new Array(); var post_id = "502830"; var post_type = "post"; var post_slug = window.location.href; var ip = "64.252.70.60"; var post_title = `Minors and Cofactors`; var post_status = "publish"; var practiceAPIURL="https://practiceapi.geeksforgeeks.org/"; var practiceURL="https://practice.geeksforgeeks.org/"; var post_date = "2021-02-19 04:14:06"; var commentSysUrl = "https://discuss.geeksforgeeks.org/commentEmbedV2.js"; //var postAdApiUrlString = ""; var link_on_code_run = ''; var link_search_modal_top = ''; var country_code_cf = "US"; var postAdApiUrlString = '4677/4685/6263/4667/4677/5967/6312/'; </script>  <meta name="description" content="Minors and Cofactors are important to calculate the adjoint and inverse of a matrix. 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category-school-learning tag-maths-class-12 tag-matrix-and-determinant" id="post-502830">  <div class="main_wrapper" style="display:flex;"> <div style="width: 100%;"> <div class="article-title"> <h1>Minors and Cofactors</h1> </div> <div class="last_updated_parent"> <div> <span class="strong">Last Updated : </span> <span>31 Jul, 2024</span> </div>  <div class="three_dot_dropdown"> <div style="display: flex;"> <div pid="502830" ptitle="Minors and Cofactors" class="top-summary-icon-div "> <div class="three_dot_dropdown_div tooltip"> <span class="summary_tooltiptext">Summarize</span> <div class="summarize_header"> <i class= "summary_button" style="position: relative;"></i> </div> </div> </div> <div pid="502830" ptitle="Minors and Cofactors" class="article--viewer_comment tooltip top-comment-icon-div"> <span class="comment_tooltiptext">Comments</span> <div class="three_dot_dropdown_div" data-gfg-action='loadComments'> <div class="comment_header"> <i class= "discussion_button" 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src=https://media.geeksforgeeks.org/auth-dashboard-uploads/Google-news.svg loading="lazy" alt="News"></img> <span class='kebab_menu_news_text'>Follow</span> </a> </div> </div> </div> <div class="text"> <p dir="ltr" style="text-align: justify;"><b><strong>Minors and Cofactors</strong></b><span> are important to calculate the adjoint and inverse of a </span><a href="https://www.geeksforgeeks.org/matrices-and-its-types-class-12-maths/" rel="noopener"><span>matrix</span></a><span>. As the name suggests, a Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs. A cofactor is (-1)</span><sup><span>i+j</span></sup><span> times the minor of the matrix.</span></p> <p dir="ltr" style="text-align: justify;"><span>They are the backbones of Linear Algebra and are used to find the value of a matrix’s determinant, adjoint, and inverse. Other than that there are many use cases in computer science for minors and cofactors. In this article, we will study minors and cofactors in detail. Other than that, we will also learn about the determinants, matrix inversion, and many more.</span></p><div id="GFG_AD_gfg_mobile_336x280" style="margin: 5px 0;"></div> <div id="table_of_content"><p style="margin:4px; font-size:20px; font-weight:bold;">Table of Content</p> <ul><li><a href="#minors-and-cofactors-of-determinants">Minors and Cofactors of Determinants</a></li><li><a href="#what-is-the-determinant-of-a-matrix">What is the Determinant of a Matrix?</a></li><li><a href="#minor-of-a-matrix">Minor of a Matrix</a></li><ul><li><a href="#how-to-find-minor-of-a-matrix">How to Find Minor of a Matrix?</a></li></ul><li><a href="#sample-problems-on-minor-of-a-matrix">Sample Problems on Minor of a Matrix</a></li><li><a href="#cofactor-of-a-matrix">Cofactor of a Matrix</a></li><li><a href="#sample-problems-on-cofactors-of-a-matrix">Sample Problems on Cofactors of a Matrix</a></li><li><a href="#application-of-minors-and-cofactors">Application of Minors and Cofactors</a></li><ul><li><a href="#adjoint-of-matrix">Adjoint of Matrix</a></li><li><a href="#inverse-of-matrix">Inverse of Matrix</a></li></ul><li><a href="#minors-and-cofactors-class-12">Minors and Cofactors Class 12</a></li><ul><li><a href="#resources-related-to-minors-and-cofactors-class-12">Resources related to Minors and Cofactors Class 12</a></li></ul></ul></div><h2 id="minors-and-cofactors-of-determinants"><span>Minors and Cofactors of Determinants</span></h2><p dir="ltr"><b><strong>Minors</strong></b><span> of a matrix are the determinants of the smaller matrices that are left after removing one row and one column from the original matrix and the cofactor of an element is calculated by multiplying the minor by</span><b><strong> (−1)</strong></b><b><sup><strong>i+j</strong></sup></b><b><strong> </strong></b><span>which accounts for the position of the element within the matrix.</span></p> <p dir="ltr"><span>To understand minors and cofactors, we should understand about determinants first.</span></p> <h2 id="what-is-the-determinant-of-a-matrix"><span>What is the Determinant of a Matrix?</span></h2><p dir="ltr" style="text-align: justify;"><a href="https://www.geeksforgeeks.org/how-to-find-the-determinant-of-a-matrix/" rel="noopener"><span>Determinant</span></a><span> are useful for solving linear equations, capturing how linear transformations change area or volume, and changing variables in integrals. The determinant can be viewed as a function whose input is a square matrix and whose output is a number. In simple language, we can say, that to every small matrix A, we can associate a number (real or complex) which is called the determinant of a square matrix A. </span></p> <p dir="ltr" style="text-align: justify;"><span>The</span><a href="https://www.geeksforgeeks.org/problems/determinant-of-a-matrix-1587115620/0" rel="noopener" target="_self"><span> determinant of a matrix </span></a><span>A can be easily represented as </span><b><strong>det(A) </strong></b><span>or </span><b><strong>|A|.</strong></b></p><div id="GFG_AD_Desktop_InContent_ATF_336x280" style="text-align:center; max-height: 280px;"></div><div id="GFG_AD_gfg_mobile_336x280_1" style="margin: 5px 0;"></div> <img alt=" Determinant of a Matrix" height="inherit" src="https://media.geeksforgeeks.org/wp-content/uploads/20230609104732/Minors-and-Cofactors-(1).png" srcset="https://media.geeksforgeeks.org/wp-content/uploads/20230609104732/Minors-and-Cofactors-(1).png 1000w, https://media.geeksforgeeks.org/wp-content/uploads/20230609104732/Minors-and-Cofactors-(1)-100.png 100w, https://media.geeksforgeeks.org/wp-content/uploads/20230609104732/Minors-and-Cofactors-(1)-200.png 200w, https://media.geeksforgeeks.org/wp-content/uploads/20230609104732/Minors-and-Cofactors-(1)-300.png 300w, https://media.geeksforgeeks.org/wp-content/uploads/20230609104732/Minors-and-Cofactors-(1)-660.png 660w, https://media.geeksforgeeks.org/wp-content/uploads/20230609104732/Minors-and-Cofactors-(1)-768.png 768w, " width="1000"/><h2 id="minor-of-a-matrix"><span>Minor of a Matrix</span></h2><p dir="ltr"><span>Minor of an element </span><b><strong>a</strong></b><b><sub><strong>ij</strong></sub></b><sub><span> </span></sub><span>of a determinant is a determinant obtained by deleting the i</span><sup><span>th </span></sup><span>row and j</span><sup><span>th </span></sup><span>column in which element </span><b><strong>a</strong></b><b><sub><strong>ij</strong></sub></b><sub><span> </span></sub><span>lies. Minor of an element </span><b><strong>a</strong></b><b><sub><strong>ij</strong></sub></b><sub><span> </span></sub><span>is denoted by </span><b><strong>M</strong></b><b><sub><strong>ij</strong></sub></b></p><div id="_GFG_ABP_Incontent_728x90" style="text-align:center;"></div> <h3 id="how-to-find-minor-of-a-matrix"><b><strong>How to Find Minor of a Matrix?</strong></b></h3><p dir="ltr"><span>Use the following steps to find the minor of any given matrix:</span></p> <blockquote><p dir="ltr"><b><strong>Step 1: </strong></b><span>Hide the i</span><sup><span>th</span></sup><span> row and j</span><sup><span>th</span></sup><span> column of the matrix A, where the element a</span><sub><span>ij</span></sub><span> lies. </span></p> <p dir="ltr"><b><strong>Step 2:</strong></b><span> Now compute the determinant of the matrix after the row and column is removed using step 1.</span></p> <p dir="ltr"><b><strong>Step 3:</strong></b><span> Result of Step 2 is the minor for he elelment of the i</span><sup><span>th</span></sup><span> row and j</span><sup><span>th</span></sup><span> column, then repeat the process for each element of the matrix to find the minor for all the elements of the matrix.</span></p> </blockquote><h2 id="sample-problems-on-minor-of-a-matrix"><span>Sample Problems on Minor of a Matrix</span></h2><p dir="ltr"><b><strong>Problem 1: If the matrix </strong></b><img src="https://quicklatex.com/cache3/80/ql_358dc301d454eccc1ea789ed0c0f6c80_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \bold{A = \begin{bmatrix} 1&2&3\\ 2&0&1\\ 5&3&6 \end{bmatrix}}    " title="Rendered by QuickLaTeX.com" height="95" width="171" style="vertical-align: 4px;"/><b><strong>then find the minor of a</strong></b><b><sub><strong>22.</strong></sub></b></p><div style="text-align:center;margin:20px 0; max-height: 280px;" id="GFG_AD_gfg_outstream_incontent"></div> <p dir="ltr"><b><strong>Solution:</strong></b></p> <blockquote><p dir="ltr"><span>In this question, we have to find out the minor of a</span><sub><span>22</span></sub><span>, the element present at a</span><sub><span>22 </span></sub><span>is 0. As we learn from our definition of a minor we have to delete the i</span><sup><span>th</span></sup><span> row and j</span><sup><span>th</span></sup><span> columns at which our asked element is present. Below image is demonstrating how to delete the i</span><sup><span>th</span></sup><span> row and j</span><sup><span>th</span></sup><span> column</span></p> <img alt="minor of a22" height="inherit" loading="lazy" src="https://media.geeksforgeeks.org/wp-content/uploads/20230518180842/Minors-and-Cofactors-01-copy.webp" srcset="https://media.geeksforgeeks.org/wp-content/uploads/20230518180842/Minors-and-Cofactors-01-copy.webp 1000w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180842/Minors-and-Cofactors-01-copy-100.webp 100w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180842/Minors-and-Cofactors-01-copy-200.webp 200w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180842/Minors-and-Cofactors-01-copy-300.webp 300w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180842/Minors-and-Cofactors-01-copy-660.webp 660w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180842/Minors-and-Cofactors-01-copy-768.webp 768w, " width="1000"/><p dir="ltr"><span>After deletion, we write our left element as it is and do cross multiplication.</span></p> <p dir="ltr"><span>Minor of element a</span><sub><span>22</span></sub><span> =</span><img src="https://quicklatex.com/cache3/f9/ql_36dcbbde93df565807878fb7c430f1f9_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \begin{vmatrix} 1&3\\ 5& 6 \end{vmatrix} " title="Rendered by QuickLaTeX.com" height="65" width="61" style="vertical-align: 8px;"/></p> <p dir="ltr"><span>⇒ Minor of element a</span><sub><span>22</span></sub><span> = 6 – 15 = -9</span></p> </blockquote><p dir="ltr"><b><strong>Note: </strong></b><span>Always remember, after the multiplication of the left diagonal element in the second-dimensional determinant, always put the -ve sign then do the multiplication of right diagonal elements and solve them out.</span></p> <p dir="ltr"><b><strong>Problem 2: If the matrix </strong></b><img src="https://quicklatex.com/cache3/47/ql_fbaecd5ab272f5929ebcf98c33a6d747_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\bold{A = \begin{bmatrix} 1 & 0 & 4\\ 3 & 5 &-1\\ 0 & 1 & 2 \end{bmatrix}}            " title="Rendered by QuickLaTeX.com" height="95" width="192" style="vertical-align: 4px;"/><b><strong>, then find the minor of a</strong></b><b><sub><strong>32</strong></sub></b><b><strong>.</strong></b></p> <p dir="ltr"><b><strong>Solution:</strong></b></p> <blockquote><p dir="ltr"><span>In the above question we have asked to find out the minor of a</span><sub><span>32</span></sub><span> element which is 1. So as we did in this above problem same procedure we will follow. So firstly we have to delete the i</span><sup><span>th</span></sup><span> row and j</span><sup><span>th</span></sup><span> column at which our element is present.</span></p> <img alt="minor of a32" height="inherit" loading="lazy" src="https://media.geeksforgeeks.org/wp-content/uploads/20230518180843/Minors-and-Cofactors-02-copy.webp" srcset="https://media.geeksforgeeks.org/wp-content/uploads/20230518180843/Minors-and-Cofactors-02-copy.webp 1000w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180843/Minors-and-Cofactors-02-copy-100.webp 100w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180843/Minors-and-Cofactors-02-copy-200.webp 200w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180843/Minors-and-Cofactors-02-copy-300.webp 300w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180843/Minors-and-Cofactors-02-copy-660.webp 660w, https://media.geeksforgeeks.org/wp-content/uploads/20230518180843/Minors-and-Cofactors-02-copy-768.webp 768w, " width="1000"/><p dir="ltr"><span>So we had canceled the i</span><sup><span>th</span></sup><span> row and j</span><sup><span>th</span></sup><span> column at which our element is present. So write the elements which are left as it is.</span></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-1"></div> <p dir="ltr"><span>Then do the cross multiplication and solve:</span></p> <p dir="ltr"><span>Minor of element a</span><sub><span>32</span></sub><span> =</span><img src="https://quicklatex.com/cache3/d1/ql_b11ce16b4f0737530f154c84b17257d1_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \begin{vmatrix} 1&4\\ 3& -1 \end{vmatrix} " title="Rendered by QuickLaTeX.com" height="65" width="82" style="vertical-align: 8px;"/></p> <p dir="ltr"><span>Minor of element a</span><sub><span>32</span></sub><span> = -1 -12 = -13</span></p> </blockquote><h2 id="cofactor-of-a-matrix"><span>Cofactor of a Matrix</span></h2><p dir="ltr"><a href="https://www.geeksforgeeks.org/cofactor-matrix/" rel="noopener" target="_blank"><span>Co-factor</span></a><span> of an element a</span><sub><span>ij </span></sub><span>of a determinant, denoted by A</span><sub><span>ij</span></sub><span> or C</span><sub><span>ij</span></sub><span>, and is defined as follows:</span></p> <blockquote><p dir="ltr" style="text-align: center;"><b><strong>C</strong></b><b><sub><strong>ij</strong></sub></b><b><strong> = (-1)</strong></b><b><sup><strong>i+j</strong></sup></b><b><strong> M</strong></b><b><sub><strong>ij</strong></sub></b><b><strong> </strong></b></p> <p dir="ltr"><span>Where,</span></p> <ul><li value="1"><b><strong>M</strong></b><b><sub><strong>ij</strong></sub></b><span> is the minor of the element </span><b><strong>a</strong></b><b><sub><strong>ij</strong></sub></b><span>, and </span></li><li value="2"><b><strong>i</strong></b><span> and </span><b><strong>j</strong></b><span> respectively reprensent the number of row and column of the element (position).</span></li></ul></blockquote><h2 id="sample-problems-on-cofactors-of-a-matrix"><span>Sample Problems on Cofactors of a Matrix</span></h2><p dir="ltr"><b><strong>Problem 1:</strong></b><span> </span><b><strong>If a matrix A is</strong></b><span> </span><img src="https://quicklatex.com/cache3/08/ql_f96fc99a0346ac55ec207be62e772908_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\bold{|A| = \begin{vmatrix} 5&3&1\\ 2&0&-1\\1&2&3 \end{vmatrix}}    " title="Rendered by QuickLaTeX.com" height="97" width="192" style="vertical-align: 4px;"/><b><strong>write the cofactor of the element a</strong></b><b><sub><strong>32.</strong></sub></b></p> <p dir="ltr"><b><strong>Solution:</strong></b></p> <blockquote><p dir="ltr"><span>As asked in question we have to find the co factor of element a32 which means our row (i) = 3 and column (j) = 2 so we have row and column as we do to find the minor by deleting the rows and column at which asked element exist we do the same in this question to and then put that in our formula -> </span><b><strong>A</strong></b><b><sub><strong>ij</strong></sub></b><b><strong> = (-1)</strong></b><b><sup><strong>i+j</strong></sup></b><b><strong> M</strong></b><b><sub><strong>ij</strong></sub></b></p> <p dir="ltr"><span>Cofactor of elements</span><b><strong> </strong></b><img src="https://quicklatex.com/cache3/e5/ql_3d38db1c7955c0712206e2ccf4d051e5_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="= (-1)^{3+2}\begin{vmatrix}5 & 1\\2&-1 \end{vmatrix}" title="Rendered by QuickLaTeX.com" height="65" width="208" style="vertical-align: -25px;"/></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-2"></div> <p dir="ltr"><span>⇒ Cofactor of element = (-1)(-5 – 2) = 7</span></p> <p dir="ltr"><span>So after putting in the formula of finding cofactor and doing expansion of determinant we get (-1) (-5 – 2) which on solving gives the answer 7, this is our required answer.</span></p> </blockquote><p dir="ltr"><b><strong>Problem 2:</strong></b><span> If A</span><sub><span>ij</span></sub><span> of the element a</span><sub><span>ij</span></sub><span> of the determinant is given below, then write the value of a</span><sub><span>32</span></sub><span> . A</span><sub><span>32</span></sub><span>.</span></p> <p dir="ltr" style="text-align: center;"><img src="https://quicklatex.com/cache3/49/ql_702f9d3200620e8d0c14b55ea9775449_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" \bold{|A| = \begin{vmatrix} 2&-3&5\\ 6&0&4\\1&5&-7 \end{vmatrix}} " title="Rendered by QuickLaTeX.com" height="97" width="212" style="vertical-align: 4px;"/></p> <p dir="ltr"><b><strong>Solution:</strong></b><span> </span></p> <blockquote><p dir="ltr"><span>In the question, we are having determinant. So we have row and column given in the question.</span></p> <p dir="ltr"><span>Here, a</span><sub><span>32</span></sub><span> = 5 (second element of third row)</span></p> <p dir="ltr"><span>Given, A</span><sub><span>ij</span></sub><span> is the cofactor of the element a</span><sub><span>ij</span></sub><span> of A . So now we can solve this question by putting the values in the formula of cofactor as discussed in above question.</span></p> <p dir="ltr"><span>Cofactor of elements</span><b><strong> </strong></b><span>(A</span><sub><span>32</span></sub><span>)</span><img src="https://quicklatex.com/cache3/59/ql_a53858732123fc937878ad2cbb307659_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="= (-1)^{3+2}\begin{vmatrix}2 &5\\6&4 \end{vmatrix} " title="Rendered by QuickLaTeX.com" height="65" width="187" style="vertical-align: 8px;"/></p> <p dir="ltr"><span>⇒ Cofactor of element (A</span><sub><span>32</span></sub><span>) = (-1)(8 – 30) = 22</span></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-3"></div> <p dir="ltr"><span> a</span><sub><span>32</span></sub><span> . A</span><sub><span>32</span></sub><span> = 5 . 22 = 110</span></p> <p dir="ltr"><span>So, 110 is our required answer.</span></p> </blockquote><h2 id="application-of-minors-and-cofactors"><span>Application of Minors and Cofactors</span></h2><p dir="ltr"><span>Minors and Cofactors are used in the calculation of the following terms:</span></p> <ul><li value="1"><a href="https://www.geeksforgeeks.org/adjoint-of-a-matrix-determinants/" rel="noopener"><span>Adjoint of Matrix</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/inverse-of-a-matrix/" rel="noopener"><span>Inverse of Matrix</span></a></li></ul><h3 id="adjoint-of-matrix"><span>Adjoint of Matrix</span></h3><p dir="ltr"><span>To calculate the adjoint of Matrix, you need to follow the following steps:</span></p> <blockquote><p dir="ltr"><b><strong>Step 1: </strong></b><span>Calculate the cofactors of each element of a given matrix.</span></p> <p dir="ltr"><b><strong>Step 2:</strong></b><span> Construct the matrix from the cofactor of elements.</span></p> <p dir="ltr"><b><strong>Step 3: </strong></b><span>Calculate the Transpose of the resultant matrix in Step 2.</span></p> <p dir="ltr"><b><strong>Step 4: </strong></b><span>Resulting matrix of Step 3 is the adjoint of the given matrix.</span></p> </blockquote><p dir="ltr"><b><strong>Example: Find the adjoint of the following matrix A;</strong></b></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/a8/ql_177bfe5dc086a3973574c4b190ca35a8_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\bold{A = \begin{bmatrix} 1&2&3\\ 4 &5 &6\\ 7&8&9 \end{bmatrix}} " title="Rendered by QuickLaTeX.com" height="95" width="171" style="vertical-align: 4px;"/></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-4"></div> <p dir="ltr"><b><strong>Solution:</strong></b></p> <blockquote><p dir="ltr"><b><strong>Step 1:</strong></b><span> Compute the cofactors of each element in A.</span></p> <p dir="ltr"><span>C</span><sub><span>11</span></sub><span> = 5 × 9 – 6 × 8 = -3</span><br/><span>C</span><sub><span>12</span></sub><span> = -(4 × 9 – 6 × 7) = 6</span><br/><span>C</span><sub><span>13</span></sub><span> = 4 × 8 – 5 × 7 = -3</span><br/><span>C</span><sub><span>21</span></sub><span> = -(2 × 9 – 3 × 8) = 6</span><br/><span>C</span><sub><span>22</span></sub><span> = 1 × 9 – 3 × 7 = -6</span><br/><span>C</span><sub><span>23</span></sub><span> = -(1 × 8 – 2 × 7) = 3</span><br/><span>C</span><sub><span>31</span></sub><span> = 2 × 6 – 3 × 5 = -3</span><br/><span>C</span><sub><span>32</span></sub><span> = -(1 × 6 – 3 × 4) = 6</span><br/><span>C</span><sub><span>33</span></sub><span> = 1 × 5 – 2 × 4 = -3</span></p> <p dir="ltr"><b><strong>Step 2:</strong></b><span> Construct the matrix of cofactors.</span></p> <p dir="ltr"><span>Matrix of cofactors, </span><img src="https://quicklatex.com/cache3/7d/ql_0603fb6e0d2277a60c4514fcf4799b7d_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="C = \begin{bmatrix} -3&6&-3\\ 6&-6&3\\ -3&6&-3 \end{bmatrix} " title="Rendered by QuickLaTeX.com" height="95" width="231" style="vertical-align: 4px;"/></p> <p dir="ltr"><b><strong>Step 3: </strong></b><span>Transpose the matrix of cofactors.</span></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/ae/ql_9a83af711e879e8330c3d56340d6b9ae_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="C' = \begin{bmatrix} -3&6&-3\\ 6&-6&6\\ -3&3&-3 \end{bmatrix} " title="Rendered by QuickLaTeX.com" height="95" width="237" style="vertical-align: 4px;"/></p> <p dir="ltr"><b><strong>Step 4: </strong></b><span>The resulting matrix is the adjoint of A.</span></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/4f/ql_e36d6782a35baa28ece394c0db546c4f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="adj(A) = C' = \begin{bmatrix} -3&6&-3\\ 6&-6&6\\ -3&3&-3 \end{bmatrix} " title="Rendered by QuickLaTeX.com" height="95" width="350" style="vertical-align: 4px;"/></p> </blockquote><h3 id="inverse-of-matrix"><span>Inverse of Matrix</span></h3><p dir="ltr"><span>To calculate the </span><a href="https://www.geeksforgeeks.org/inverse-of-matrix/" rel="noreferrer noopener" target="_self"><span>inverse of a matrix</span></a><span>, you can use the following steps:</span></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-5"></div> <blockquote><p dir="ltr"><b><strong>Step 1: </strong></b><span>Calculate the determinant of the given matrix.</span></p> <p dir="ltr"><b><strong>Step 2:</strong></b><span> If the value of the determinant is zero, then the matrix has no inverse. Otherwise, calculate the adjoint of the matrix using the steps mentioned in the previous heading.</span></p> <p dir="ltr"><b><strong>Step 3: </strong></b><span>Use the formula </span><img src="https://quicklatex.com/cache3/61/ql_220c99f43795a10c1098b24376386661_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="A^{-1} = \frac{adj(A)}{|A|} " title="Rendered by QuickLaTeX.com" height="43" width="145" style="vertical-align: 18px;"/><span>, to calculate the inverse of the given matrix.</span></p> </blockquote><p dir="ltr"><b><strong>Example: Find the inverse of the following matrix A:</strong></b></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/9c/ql_1cd281cfaf98bd6c08279e13b1610c9c_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\bold{A = \begin{bmatrix} 2&-1&0\\ -1 &2 &-1\\ 0&-1&2 \end{bmatrix}} " title="Rendered by QuickLaTeX.com" height="95" width="233" style="vertical-align: 4px;"/></p> <p dir="ltr"><b><strong>Solution:</strong></b></p> <blockquote><p dir="ltr"><b><strong>Step 1:</strong></b><span> Find the determinant of A.</span></p> <p dir="ltr"><span>det(A) = 2(2 × 2 – (-1) × (-1)) – (-1)((-1) ×2 – (-1) ×0) + 0 </span></p> <p dir="ltr"><span>⇒ det(A) = 6 – 2 = 4</span></p> <p dir="ltr"><b><strong>Step 2: </strong></b><span>Calculate the adjoint of A using the steps mentioned in the previous example.</span></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-6"></div> <p dir="ltr"><span>C</span><sub><span>11</span></sub><span> = 4 – 1 = 3</span><br/><span>C</span><sub><span>12</span></sub><span> = -(-2 – 0) = 2</span><br/><span>C</span><sub><span>13</span></sub><span> = 1 – 0 = 1</span><br/><span>C</span><sub><span>21</span></sub><span> = -(-2 – 0) = 2</span><br/><span>C</span><sub><span>22</span></sub><span> = 4 – 0 = 4</span><br/><span>C</span><sub><span>23</span></sub><span> = -(-2 – 0) = 2</span><br/><span>C</span><sub><span>31</span></sub><span> = (1 – 0) = 1</span><br/><span>C</span><sub><span>32</span></sub><span> = -(-2 – 0) = 2</span><br/><span>C</span><sub><span>33</span></sub><span> = 4 – 1 = 3</span></p> <p dir="ltr"><span>Matrix of cofactors,</span></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/07/ql_9ca3b2a33a3686e8812594114f65e907_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="C = \begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix} " title="Rendered by QuickLaTeX.com" height="95" width="169" style="vertical-align: 4px;"/></p> <p dir="ltr"><span>Transpose of matrix of cofactors,</span></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/d4/ql_e9ad4b47d8ef88003637611ef95dddd4_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="adj(A) = C' =  \begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix} " title="Rendered by QuickLaTeX.com" height="95" width="288" style="vertical-align: 4px;"/></p> <p dir="ltr"><b><strong>Step 3: </strong></b><span>Multiply the adjoint of A by the reciprocal of the determinant.</span></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/68/ql_ca10c999a548c6b74b9db761cf93b668_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt=" A^{-1} = \frac{adj(A)}{4} = \frac{1}{4} \times\begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix} " title="Rendered by QuickLaTeX.com" height="95" width="345" style="vertical-align: 4px;"/></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/65/ql_50eedd87372cc327536bffba2e6a4365_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Rightarrow A^{-1}= \begin{bmatrix} \frac{3}{4}&\frac{2}{4}&\frac{1}{4}\\ \frac{2}{4}&\frac{4}{4}&\frac{2}{4}\\ \frac{1}{4}&\frac{2}{4}&\frac{3}{4} \end{bmatrix} " title="Rendered by QuickLaTeX.com" height="98" width="238" style="vertical-align: 3px;"/></p> <p dir="ltr"><img src="https://quicklatex.com/cache3/dd/ql_c80111408e6944b5c3e8b1d7d7b684dd_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="\Rightarrow A^{-1}= \begin{bmatrix} \frac{3}{4}&\frac{1}{2}&\frac{1}{4}\\ \frac{1}{2}&1&\frac{1}{2}\\ \frac{1}{4}&\frac{1}{2}&\frac{3}{4} \end{bmatrix} " title="Rendered by QuickLaTeX.com" height="98" width="238" style="vertical-align: 3px;"/></p> </blockquote><h3 id="people-also-read"><b><strong>People Also Read:</strong></b></h3><blockquote><ul><li value="1"><a href="https://www.geeksforgeeks.org/elementary-operations-on-matrices/" rel="noopener"><span>Elementary Operation on Matrices</span></a></li><li value="2"><a href="https://www.geeksforgeeks.org/order-of-matrix/" rel="noopener"><span>Order of Matrix</span></a></li></ul></blockquote><h2 id="minors-and-cofactors-class-12"><span>Minors and Cofactors Class 12</span></h2><p dir="ltr"><b><strong>Minors</strong></b><span> and cofactors of a matrix are important concepts used in the calculation of determinants and in solving systems of linear equations especially for class 12. </span></p> <h3 id="resources-related-to-minors-and-cofactors-class-12"><span>Resources related to Minors and Cofactors Class 12</span></h3><ul><li value="1"><a href="https://" rel="noopener noreferrer"><span>Matrices and Determinants Class 12 Notes </span></a></li><li value="2"><span>Matrices and Determinants Class 12 NCERT Soutions</span></li></ul><h2 id="faqs-on-minors-and-cofactors"><span>FAQs on Minors and Cofactors</span></h2><h3 id="what-is-a-minor-in-matrix-algebra"><span>What is a Minor in Matrix Algebra?</span></h3><blockquote><p dir="ltr"><span> In matrix algebra, a minor refers to the determinant of a smaller matrix obtained by deleting one or more rows and columns from a larger matrix.</span></p> </blockquote><h3 id="what-is-a-cofactor-in-matrix-algebra"><span>What is a Cofactor in Matrix Algebra?</span></h3><blockquote><p dir="ltr"><span>In matrix algebra, the cofactor of an element in a matrix is defined as the signed minor obtained by deleting the row and column containing that element, and multiplying the resulting determinant by -1 if the sum of the row and column indices is odd.</span></p> </blockquote><h3 id="what-is-the-relationship-between-minors-and-cofactors"><span>What is the Relationship between Minors and Cofactors?</span></h3><blockquote><p dir="ltr"><span>The cofactor of an element in a matrix is equal to the product of the minor of that element and (-1)^(i+j), where i and j are the row and column indices of the element, respectively.</span></p> </blockquote><h3 id="what-is-the-use-of-minors-and-cofactors-in-matrix"><span>What is the use of Minors and Cofactors in Matrix?</span></h3><blockquote><p dir="ltr"><span>Minors and cofactors are used in the computation of the inverse of a matrix, as well as in the calculation of determinants and solutions to systems of linear equations.</span></p> </blockquote><h3 id="how-are-minors-and-cofactors-used-to-calculate-the-inverse-of-a-matrix"><span>How are Minors and Cofactors used to calculate the Inverse of a Matrix?</span></h3><blockquote><p dir="ltr"><span>To calculate the inverse of a matrix we first need to calculate the cofactor matrix of any given matrix, then transpose of cofactor matrix is called adjoint. Then dividing adjoint by determinant to get the inverse of the matrix.</span></p> </blockquote><h3 id="can-minors-and-cofactors-be-used-to-solve-systems-of-linear-equations"><span>Can Minors and Cofactors be used to solve systems of Linear Equations?</span></h3><blockquote><p dir="ltr"><span>Yes, minors and cofactors can be used to solve systems of linear equations by representing the system in the form of a matrix and using Cramer’s rule, which expresses the solution in terms of determinants of smaller matrices obtained by replacing one column of the coefficient matrix with the constant terms of the system.</span></p> </blockquote><h3 id="what-is-laplaces-expansion-formulatransformations"><span>What is Laplace’s Expansion Formula?</span><b><strong>transformations</strong></b></h3><blockquote><p dir="ltr"><span>Laplace’s expansion formula is a method for computing the determinant of a matrix by summing the products of the elements in any row or column with their corresponding cofactors. It is given as </span><img src="https://quicklatex.com/cache3/f1/ql_1067eaf04d44b2172e27494734bdf4f1_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="det(A) = \sum_{j =1}^{n}(-1)^{i+j}b_{i,j}m_{i,j} " title="Rendered by QuickLaTeX.com" height="34" width="341" style="vertical-align: 21px;"/></p> </blockquote><br/><div id="AP_G4GR_6"></div> <div class='article_bottom_text'></div><br> <div id="video-tab-content" class="video-tab-content"> <div style="text-align: center; margin: 20px 0px;" id="GFG_AD_InContent_Desktop_728x280"></div> </div> <div class="d-row content-bw article-pgnavi v-divider-gfg"style="margin-top: 20px;"> <div class="article-pgnavi_prev"><div class="comment_div" data-title="Comments"><button class="author_footer_btn" data-gfg-action="loadComments"> <div class='discussion_panel'> <i class='discussion_button'> </i> Comment </div></button></div><div class="trigger-div"><button class="author_footer_btn"><div class="more_info">More info<span class="more_info_uparrow gfg-icon gfg-icon_arrow-down-thin"></span></div></button></div></div> <div class="article-pgnavi_next"> <a href="https://www.geeksforgeeks.org/adjoint-of-matrix/?ref=next_article" class="pg-head"> <span>Next Article</span> <span class="gfg-icon gfg-icon_next"></span> </a>  <div class="pg-main"> <a href="https://www.geeksforgeeks.org/adjoint-of-matrix/?ref=next_article">Adjoint of a Matrix</a> </div> </div> </div> <div class="article-meta-author-details hidden-div"> <div class="article-meta-author-details-block"> <div class="article-meta-author-details-profile-display"> <div class="author_info"> <div class="article-meta-author-details-profile-display-icon"> <a href="https://www.geeksforgeeks.org/user/srishivansh5404/contributions/?itm_source=geeksforgeeks&itm_medium=article_author&itm_campaign=auth_user" style="text-decoration: none;"><div class="image-wrap" style="position: relative"><img src="https://media.geeksforgeeks.org/auth/profile/ra9opczj2byt7zs611ak" class="avatarprofile_image restrict-popup-gfg" alt="author"> <div style="position: absolute;top: -35px; 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We define invertib</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown" style="background-color:var(--leftbar-background-color); border-bottom:1px solid #CCC;"><h2 class="dropdown-title active">Chapter 4: Determinants<span class="gfg-icon gfg-icon_arrow-down-thin upside"></span></h2><ul><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-is-determinant-of-a-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Determinant of Matrix with Solved Examples </div> <div class="gfg-similar-read-item-subheading">Determinant of a matrix is a special number that can be calculated for a square matrix (a matrix with the same number of rows and columns). It is a function that gives the unique output (real number) for every input value of the square matrix. The determinant of the matrix is considered the scaling</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/properties-of-determinants-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Properties of Determinants </div> <div class="gfg-similar-read-item-subheading">Properties of Determinants are the properties that are required to solve various problems in Matrices. There are various properties of the determinant that are based on the elements, rows, and columns of the determinant. These properties help us to easily find the value of the determinant. Suppose w</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/area-of-triangle-using-determinants/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Area of a Triangle using Determinants </div> <div class="gfg-similar-read-item-subheading">Area of a Triangle using Determinants is one of the multiple methods available to find the area of a triangle and this method is often overlooked for its complexity. But the Area of a Triangle using Determinants comes very handy in coordinate geometry when all three vertices of a triangle are given</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="currentpage similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/minors-and-cofactors/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Minors and Cofactors </div> <div class="gfg-similar-read-item-subheading">Minors and Cofactors are important to calculate the adjoint and inverse of a matrix. As the name suggests, a Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs. A cofactor is (-1)i+j ti</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/adjoint-of-matrix/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Adjoint of a Matrix </div> <div class="gfg-similar-read-item-subheading">Knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. From matrices there come Determinants, Now we see one of the properties of the Determinant in this article. In this article, we see how to find the Adjoint of a Matrix.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/applications-of-matrices-and-determinants/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Applications of Matrices and Determinants </div> <div class="gfg-similar-read-item-subheading">Applications of Matrices and Determinants: One application of matrices and determinants is that they can be used to solve linear equations in two or three variables. Matrices and determinants are also used to check the consistency of any system, whether they are consistent or not. This is the most u</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 5: Continuity and Differentiability<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/continuity-and-discontinuity-in-calculus-class-12-cbse/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Continuity and Discontinuity in Calculus </div> <div class="gfg-similar-read-item-subheading">Continuity and Discontinuity: Continuity and discontinuity are fundamental concepts in calculus and mathematical analysis, describing the behavior of functions. A function is continuous at a point if you can draw the graph of the function at that point without lifting your pen from the paper. Contin</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/differentiability-of-a-function-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Differentiability of a Function | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">Continuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-inverse-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Inverse Functions </div> <div class="gfg-similar-read-item-subheading">In mathematics, a function(e.g. f), is said to be an inverse of another(e.g. g), if given the output of g returns the input value given to f. Additionally, this must hold true for every element in the domain co-domain(range) of g. E.g. assuming x and y are constants if g(x) = y and f(y) = x then the</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-implicit-functions-continuity-and-differentiability-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Implicit Functions - Continuity and Differentiability | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">Implicit functions are functions where a specific variable cannot be expressed as a function of the other variable. A function that depends on more than one variable. Implicit Differentiation helps us compute the derivative of y with respect to x without solving the given equation for y, this can be</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-composite-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Composite Functions </div> <div class="gfg-similar-read-item-subheading">Derivatives are an essential part of calculus. They help us in calculating the rate of change, maxima, and minima for the functions. Derivatives by definition are given by using limits, which is called the first form of the derivative. We already know how to calculate the derivatives for standard fu</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-inverse-trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Inverse Trigonometric Functions </div> <div class="gfg-similar-read-item-subheading">Derivatives of Inverse Trigonometric Functions: Every mathematical function, from the simplest to the most complex, has an inverse. In mathematics, the inverse usually means the opposite. In addition, the inverse is subtraction. For multiplication, it's division. In the same way for trigonometric fu</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivative-of-exponential-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivative of Exponential Functions </div> <div class="gfg-similar-read-item-subheading">Derivative of Exponential Function stands for differentiating functions expressed in the form of exponents. We know that exponential functions exist in two forms, ax where a is a real number r and is greater than 0 and the other form is ex where e is Euler's Number and the value of e is 2.718 . . .</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/logarithmic-differentiation-continuity-and-differentiability/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Logarithmic Differentiation - Continuity and Differentiability </div> <div class="gfg-similar-read-item-subheading">The word continuity means something which is continuous in nature. The flow of water is continuous, time in real life is continuous, and many more instances show the continuity in real life. In mathematics, the Continuous function is the one which when drawn on a graph does not show any breaks and i</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/proofs-for-the-derivatives-of-e%cb%a3-and-lnx-advanced-differentiation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Proofs for the derivatives of eˣ and ln(x) - Advanced differentiation </div> <div class="gfg-similar-read-item-subheading">In this article, we are going to cover the proofs of the derivative of the functions ln(x) and ex. Before proceeding there are two things that we need to revise: The first principle of derivative Finding the derivative of a function by computing this limit is known as differentiation from first prin</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">3 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/rolles-theorem-and-lagranges-mean-value-theorem/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Rolle's Theorem and Lagrange's Mean Value Theorem </div> <div class="gfg-similar-read-item-subheading">Rolle's Theorem and Lagrange's Mean Value Theorem: Mean Value Theorems (MVT) are the basic theorems used in mathematics. They are used to solve various types of problems in Mathematics. Mean Value Theorem is also called Lagrenges's Mean Value Theorem. Rolle’s Theorem is a subcase of the mean value t</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivative-of-functions-in-parametric-form/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivative of Functions in Parametric Forms </div> <div class="gfg-similar-read-item-subheading">Parametric Differentiation refers to the differentiation of a function in which the dependent and independent variables are equated to a third variable. Derivatives of the functions express the rate of change in the functions. We know how to calculate the derivatives for standard functions. Chain ru</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/second-order-derivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Second Order Derivatives: Rules , Formula and Examples (Class 12 Maths) </div> <div class="gfg-similar-read-item-subheading">The Second Order Derivative is defined as the derivative of the first derivative of the given function. The first-order derivative at a given point gives us the information about the slope of the tangent at that point or the instantaneous rate of change of a function at that point. Second-Order Deri</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/mean-value-theorem/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Mean Value Theorem </div> <div class="gfg-similar-read-item-subheading">The Mean Value Theorem states that for a curve passing through two given points there exists at least one point on the curve where the tangent is parallel to the secant passing through the two given points. Mean Value Theorem is abbreviated as MVT. This theorem was first proposed by an Indian Mathem</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/algebra-of-continuous-functions-continuity-and-differentiability-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Algebra of Continuous Functions - Continuity and Differentiability | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">Algebra of Continuous Functions deals with the utilization of continuous functions in equations involving the varied binary operations you've got studied so. We'll also mention a composition rule that may not be familiar to you but is extremely important for future applications. Since the continuity</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 6: Applications of Derivatives<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/critical-points/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Critical Points </div> <div class="gfg-similar-read-item-subheading">As the complexity of the functions increase, we see more and more complex behavior from their graphs, and it becomes harder to graph. There have lots of peaks and valleys in their graphs. It becomes essential to find out the position of these valleys and peaks, the peaks are called maxima and the va</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-as-rate-of-change/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives as Rate of Change </div> <div class="gfg-similar-read-item-subheading">Derivatives are a mathematical tool used to analyze how quantities change. We can calculate derivatives for various, quotient, and chain rulesfunctions, including trigonometric, exponential, polynomial, and implicit functions. There are two main methods for calculating derivatives: using limits or a</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/increasing-and-decreasing-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Increasing and Decreasing Functions </div> <div class="gfg-similar-read-item-subheading">If you're studying calculus, then you're probably familiar with the concepts of increasing and decreasing functions. These terms refer to the behavior of a function as its input values change. Specifically, an increasing function is one that becomes larger as its input values increase, while a decre</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/increasing-and-decreasing-intervals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Increasing and Decreasing Intervals </div> <div class="gfg-similar-read-item-subheading">Increasing and decreasing intervals are the intervals of real numbers in which real-valued functions are increasing and decreasing respectively. Derivatives are a way of measuring the rate of change of a variable. When it comes to functions and calculus, derivatives give us a lot of information abou</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/tangents-and-normals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Tangents and Normals </div> <div class="gfg-similar-read-item-subheading">Tangent and Normals are the lines that are used to define various properties of the curves. We define tangent as the line which touches the circle only at one point and normal is the line that is perpendicular to the tangent at the point of tangency. Any tangent of the curve passing through the poin</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/equation-of-tangents-and-normals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Equation of Tangents and Normals </div> <div class="gfg-similar-read-item-subheading">Derivatives are used to find rate of change of a function with respect to variables. To find rate of change of function with respect to a variable differentiating it with respect to that variable is required. Rate of change of function y = f(x) with respect to x is defined by dy/dx or f'(x). For exa</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/relative-minima-and-maxima/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Relative Minima and Maxima </div> <div class="gfg-similar-read-item-subheading">Relative maxima and minima are the points defined in any function such that at these points the value of the function is either maximum or minimum in their neighborhood. Relative maxima and minima depend on their neighborhood point and are calculated accordingly. We find the relative maxima and mini</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/absolute-minima-and-maxima/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Absolute Minima and Maxima </div> <div class="gfg-similar-read-item-subheading">Absolute Maxima and Minima are the maximum and minimum values of the function defined on a fixed interval. A function in general can have high values or low values as we move along the function. The maximum value of the function in any interval is called the maxima and the minimum value of the funct</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/concave-function/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Concave Function </div> <div class="gfg-similar-read-item-subheading">Graphs of the functions give us a lot of information about the nature of the function, the trends, and the critical points like maxima and minima of the function. Derivatives allow us to mathematically analyze these functions and their sign can give us information about the maximum and minimum of th</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/inflection-point/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Inflection Point </div> <div class="gfg-similar-read-item-subheading">Inflection Point describes a point where the curvature of a curve changes direction. It represents the transition from a concave to a convex shape or vice versa. Let's learn about Inflection Points in detail, including Concavity of Function and solved examples. Table of Content Inflection Point Defi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/curve-sketching/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Curve Sketching </div> <div class="gfg-similar-read-item-subheading">Curve Sketching as its name suggests helps us sketch the approximate graph of any given function which can further help us visualize the shape and behavior of a function graphically. Curve sketching isn't any sure-shot algorithm that after application spits out the graph of any desired function but</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/approximations-maxima-and-minima-application-of-derivatives-class-12-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Approximations & Maxima and Minima - Application of Derivatives | Class 12 Maths </div> <div class="gfg-similar-read-item-subheading">An approximation is similar but not exactly equal to something else. Approximation occurs when an exact numerical number is unknown or difficult to obtain. In Mathematics, we use differentiation to find the approximate values of certain quantities. Let f be a given function and let y = f(x). Let ∆x</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/higher-order-derivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Higher Order Derivatives </div> <div class="gfg-similar-read-item-subheading">Higher Order Derivatives are the second, third, or further derivative of the function, i.e. differentiating a function multiple times results in a higher order derivative. Suppose we have a function f(x) then its differentiation is f'(x) which is a first-order derivative. Then its differentiation ag</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 7: Integrals<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integrals </div> <div class="gfg-similar-read-item-subheading">Integrals: An integral in mathematics is a continuous analog of a sum that is used to determine areas, volumes, and their generalizations. Performing integration is the process of computing an integral and is one of the two basic concepts of calculus. Integral in Calculus is the branch of Mathematic</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-by-substitution/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration by Substitution Method </div> <div class="gfg-similar-read-item-subheading">Integration by substitution is one of the important methods for finding the integration of the function where direct integration can not be easily found. This method is very useful in finding the integration of complex functions. We use integration by substitution to reduce the given function into t</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-by-partial-fractions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration by Partial Fractions </div> <div class="gfg-similar-read-item-subheading">Integration by Partial Fractions is one of the methods of integration, which is used to find the integral of the rational functions. In Partial Fraction decomposition, an improper-looking rational function is decomposed into the sum of various proper rational functions. If f(x) and g(x) are polynomi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-by-parts/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration by Parts </div> <div class="gfg-similar-read-item-subheading">Integration by Parts: Integration by parts is a technique used in calculus to find the integral of the product of two functions. It's essentially a reversal of the product rule for differentiation. Integrating a function is not always easy sometimes we have to integrate a function that is the multip</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-of-trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration of Trigonometric Functions </div> <div class="gfg-similar-read-item-subheading">Integration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below. Below is an example of the Integr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/functions-defined-by-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Functions Defined by Integrals </div> <div class="gfg-similar-read-item-subheading">While thinking about functions, we always imagine that a function is a mathematical machine that gives us an output for any input we give. It is usually thought of in terms of mathematical expressions like squares, exponential and trigonometric function, etc. It is also possible to define the functi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/definite-integral/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Definite Integral | Definition, Formula & How to Calculate </div> <div class="gfg-similar-read-item-subheading">A definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/computing-definite-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Computing Definite Integrals </div> <div class="gfg-similar-read-item-subheading">Integrals are a very important part of the calculus. They allow us to calculate the anti-derivatives, that is given a function's derivative, integrals give the function as output. Other important applications of integrals include calculating the area under the curve, the volume enclosed by a surface</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/fundamental-theorem-of-calculus/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Fundamental Theorem of Calculus | Part 1, Part 2 </div> <div class="gfg-similar-read-item-subheading">Fundamental Theorem of Calculus is the basic theorem that is widely used for defining a relation between integrating a function of differentiating a function. The fundamental theorem of calculus is widely useful for solving various differential and integral problems and making the solution easy for</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/finding-derivative-with-fundamental-theorem-of-calculus/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Finding Derivative with Fundamental Theorem of Calculus </div> <div class="gfg-similar-read-item-subheading">Integrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/evaluating-definite-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Evaluating Definite Integrals </div> <div class="gfg-similar-read-item-subheading">Integration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/properties-of-definite-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Properties of Definite Integrals </div> <div class="gfg-similar-read-item-subheading">Properties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as [Tex]\int_{a}^{b}[/Tex]f(x) = F(b) − F(a) There are many properties regarding definite integral. We will discuss each property one by one with</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/definite-integrals-of-piecewise-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Definite Integrals of Piecewise Functions </div> <div class="gfg-similar-read-item-subheading">Imagine a graph with a function drawn on it, it can be a straight line or a curve or anything as long as it is a function. Now, this is just one function on the graph, can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight line</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/improper-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Improper Integrals </div> <div class="gfg-similar-read-item-subheading">Geometrically speaking, integrals are the way to compute the area or volume under curves. These methods allow mathematicians to compute the area under arbitrarily complex curves. These types of integrals are called definite integrals. Definite integrals are built upon the idea of indefinite integral</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/riemann-sum/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Riemann Sums </div> <div class="gfg-similar-read-item-subheading">Riemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/riemann-sums-in-summation-notation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Riemann Sums in Summation Notation </div> <div class="gfg-similar-read-item-subheading">Riemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/trapezoidal-rule/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Trapezoidal Rule </div> <div class="gfg-similar-read-item-subheading">The trapezoidal rule is one of the fundamental rules of integration which is used to define the basic definition of integration. It is a widely used rule and the Trapezoidal rule is named so because it gives the area under the curve by dividing the curve into small trapezoids instead of rectangles.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/definite-integral-as-the-limit-of-a-riemann-sum/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Definite Integral as the Limit of a Riemann Sum </div> <div class="gfg-similar-read-item-subheading">Definite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/antiderivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Antiderivatives </div> <div class="gfg-similar-read-item-subheading">Antiderivatives: The Antiderivative of a function is the inverse of the derivative of the function. Antiderivative is also called the Integral of a function. Suppose the derivative of a function d/dx[f(x)] is F(x) + C then the antiderivative of [F(x) + C] dx of the F(x) + C is f(x). An example expla</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/indefinite-integrals/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration Formulas </div> <div class="gfg-similar-read-item-subheading">Indefinite Integrals: The derivatives have been really useful in almost every aspect of life. They allow for finding the rate of change of a function. Sometimes there are situations where the derivative of a function is available, and the goal is to calculate the actual function whose derivative is</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/particular-solutions-to-differential-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Particular Solutions to Differential Equations </div> <div class="gfg-similar-read-item-subheading">Indefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/integration-by-u-substitution/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Integration by U-substitution </div> <div class="gfg-similar-read-item-subheading">Finding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/reverse-chain-rule/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Reverse Chain Rule </div> <div class="gfg-similar-read-item-subheading">Integrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/partial-fraction-expansion/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Partial Fraction Expansion </div> <div class="gfg-similar-read-item-subheading">If f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/trigonometric-substitution/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Trigonometric Substitution: Method, Formula and Solved Examples </div> <div class="gfg-similar-read-item-subheading">Trigonometric Substitution is one of the substitution methods of integration where a function or expression in the given integral is substituted with trigonometric functions such as sin, cos, tan, etc. Integration by substitution is the easiest substitution method. 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The areas for the sta</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 9: Differential Equations<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/differential-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Differential Equations </div> <div class="gfg-similar-read-item-subheading">A differential equation is a mathematical equation that relates a function with its derivatives. 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Table of Content Implicit DifferentiationMethod to solveImplicit differentiation Formula Solve</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/advanced-differentiation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Advanced Differentiation </div> <div class="gfg-similar-read-item-subheading">Derivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. 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} .popup-main .close-icon { background-image: var(--com-extra-icons-mobile-image);background-position: -15px -1651px;height: 22px;width: 22px;transform: scale(1); margin-bottom: 12px; cursor: pointer; } .popup-main .experience-card { color: #000; display: grid; grid-template-columns: repeat(2, 1fr); gap: 16px; /* margin-bottom:10px; */ } .popup-main a:hover, .popup-main a:active, .popup-main a:visited { color: inherit; text-decoration: none; } .popup-main .exp-card1, .popup-main .exp-card2, .popup-main .exp-card3, .popup-main .exp-card4, .popup-main .exp-card5, .popup-main .exp-card6 { display: flex; align-items: center; border-radius: 10px; cursor: pointer; } .popup-main .exp-card1 .icon1, .popup-main .exp-card2 .icon2, .popup-main .exp-card3 .icon3, .popup-main .exp-card4 .icon4, .popup-main .exp-card5 .icon5, .popup-main .exp-card6 .icon6 { width: 80px; height: 80px; border-radius: 10px 0px 0px 10px; display: flex; align-items: center; justify-content: center; background: var(--write-experience-card-icon-1); } .popup-main .exp-card1 .icon1, .popup-main .exp-card2 .icon2, .popup-main .exp-card3 .icon3, .popup-main .exp-card4 .icon4, .popup-main .exp-card5 .icon5, .popup-main .exp-card6 .icon6 { width: 80px; height: 80px; border-radius: 10px 0px 0px 10px; display: flex; align-items: center; justify-content: center; } .popup-main .exp-card1 .icon1 { background: var(--write-experience-card-icon-1); } .popup-main .exp-card2 .icon2 { background: var(--write-experience-card-icon-2); } .popup-main .exp-card3 .icon3 { background: var(--write-experience-card-icon-3); } .popup-main .exp-card4 .icon4 { background: var(--write-experience-card-icon-4); } .popup-main .exp-card5 .icon5 { background: var(--write-experience-card-icon-5); } .popup-main .exp-card6 .icon6 { background: var(--write-experience-card-icon-6); } .popup-main .exp-card6 .icon6-image { background-image: url(https://media.geeksforgeeks.org/auth-dashboard-uploads/compass.svg); width: 46px; height: 30px; background-size: 30px; background-repeat: no-repeat; background-position: center; } .popup-main .exp-card1 .icon1-image { background-image: var(--com-extra-icons-mobile-image); width: 46px; height: 28px; background-position: -4px -812px; } .popup-main .exp-card2 .icon2-image { background-image: var(--com-extra-icons-mobile-image); width: 46px; height: 28px; background-position: -4px -888px; } .popup-main .exp-card3 .icon3-image { background-image: var(--com-extra-icons-mobile-image); width: 46px; height: 28px; background-position: -4px -848px; } .popup-main .exp-card4 .icon4-image { background-image: url(https://media.geeksforgeeks.org/auth-dashboard-uploads/competitive.svg); width: 46px; height: 30px; background-size: 30px; background-repeat: no-repeat; background-position: center; } .popup-main .exp-card5 .icon5-image { background-image: var(--com-extra-icons-mobile-image); width: 46px; height: 33px; background-position: -4px -1217px; } .popup-main .exp-card1 .exp-card1-text, .popup-main .exp-card2 .exp-card2-text, .popup-main .exp-card3 .exp-card3-text, .popup-main .exp-card4 .exp-card4-text, .popup-main .exp-card5 .exp-card5-text, .popup-main .exp-card6 .exp-card6-text { display: flex; align-items: center; border-radius: 0px 10px 10px 0px; height: 80px; width: calc(100% - 80px); } .popup-main .exp-card1 { justify-content: flex-start; background: var(--write-experience-card-1); } .popup-main .exp-card2 { justify-content: flex-start; background: var(--write-experience-card-2); } .popup-main .exp-card3 { justify-content: flex-start; background: var(--write-experience-card-3); } .popup-main .exp-card4 { justify-content: flex-start; background: var(--write-experience-card-4); } .popup-main .exp-card5 { justify-content: flex-start; background: var(--write-experience-card-5); } .popup-main .exp-card6 { justify-content: flex-start; background: var(--write-experience-card-6); } .popup-main span { font-family: var(--font-primary); font-size: 14px; font-style: normal; font-weight: 500; line-height: normal; padding-left: 5px; padding-right: 5px; } #popup { display: none; background-color: var(--write-modal-background); padding: 20px; text-align: center; position: fixed; top: 50%; left: 50%; transform: translate(-50%, -50%); z-index: 1026; width: min(650px, calc(100% - 20px)); } .link-container-write{ display:flex; flex-direction:column; } .link-container-write > a { margin:7px 0px; } .phrase__container{ color:var(--color-gfg); border-radius: 6px; width: 100%; background-color:var(--write-redirect-container); padding: 15px; text-align: justify; font-size:14px; } .phrase__container:hover { color:var(--color-gfg) !important; background-color:var(--write-redirect-container-hover); } </style> <div class="popup-container"> <div id="popup" accesskey="" class="popup-main"> <div class="popup-heading"> <h2>What kind of Experience do you want to share?</h2> <div class="close-icon share-experience-modal-close"></div> </div>  <div class="experience-card"> <a class="exp-card5" href= "https://write.geeksforgeeks.org/posts-new?cid=e8fc46fe-75e7-4a4b-be3c-0c862d655ed0" target="_blank"> <div class="icon5"> <div class="icon5-image"></div> </div> <div class="exp-card5-text"> <span style="color: #000;">Interview Experiences</span> </div> </a> <a class="exp-card1" href="https://write.geeksforgeeks.org/posts-new?cid=82536bdb-84e6-4661-87c3-e77c3ac04ede" target="_blank"> <div class="icon1"> <div class="icon1-image"></div> </div> <div class="exp-card1-text"> <span style="color: #000;">Admission Experiences</span> </div> </a> <a class="exp-card6" href= "https://write.geeksforgeeks.org/posts-new?cid=5219b0b2-7671-40a0-9bda-503e28a61c31" target="_blank"> <div class="icon6"> <div class="icon6-image"></div> </div> <div class="exp-card6-text"> <span style="color: #000;">Career Journeys</span> </div> </a> <a class="exp-card2" href="https://write.geeksforgeeks.org/posts-new?cid=22ae3354-15b6-4dd4-a5b4-5c7a105b8a8f" target="_blank"> <div class="icon2"> <div class="icon2-image"></div> </div> <div class="exp-card2-text"> <span style="color: #000;">Work Experiences</span> </div> </a> <a class="exp-card3" href= "https://write.geeksforgeeks.org/posts-new?cid=c5e1ac90-9490-440a-a5fa-6180c87ab8ae" target="_blank"> <div class="icon3"> <div class="icon3-image"></div> </div> <div class="exp-card3-text"> <span style="color: #000;">Campus Experiences</span> </div> </a> <a class="exp-card4" href= "https://write.geeksforgeeks.org/posts-new?cid=5ebb8fe9-b980-4891-af07-f2d62a9735f2" target="_blank"> <div class="icon4"> <div class="icon4-image"></div> </div> <div class="exp-card4-text"> <span style="color: #000;">Competitive Exam Experiences</span> </div> </a> </div>  </div> <div id="overlay" onclick="toggleExperiencePopup()" style="display: block;min-height: 100vh;min-width: 100vw;position: fixed;top: 0;bottom: 0;right: 0;left: 0;display: none;justify-content: center;align-items: center;background: rgba(0,0,0,.702);z-index: 1025 !important;backdrop-filter: blur(4px); -webkit-backdrop-filter: blur(4px);word-break: keep-all;"></div> </div> <script> $('.share-experience-modal').click(function(e){ e.preventDefault(); var link = $(this).attr('href'); toggleExperiencePopup(link); }); function toggleExperiencePopup(link) { var popup = document.getElementById("popup"); var overlay = document.getElementById("overlay"); if (window.innerWidth < 992) { window.location.href = link; } else { popup.style.display = (popup.style.display === "block") ? "none" : "block"; overlay.style.display = (overlay.style.display === "block") ? "none" : "block"; } } $('.share-experience-modal-close').click(function(e){ var popup = document.getElementById("popup"); popup.style.display = 'none'; var overlay = document.getElementById("overlay"); overlay.style.display = 'none'; }); </script> </div><script type="text/javascript"> $(window).on('load', function() { (function(c,l,a,r,i,t,y){ c[a]=c[a]||function(){(c[a].q=c[a].q||[]).push(arguments)}; t=l.createElement(r);t.async=1;t.src="https://www.clarity.ms/tag/"+i; y=l.getElementsByTagName(r)[0];y.parentNode.insertBefore(t,y); })(window, document, "clarity", "script", "ayc7ypwwuk"); }); </script>  </body> </html>

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