{"id":5139,"date":"2021-09-08T20:44:55","date_gmt":"2021-09-08T15:14:55","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5139"},"modified":"2021-11-22T20:40:09","modified_gmt":"2021-11-22T15:10:09","slug":"differentiation-of-determinant","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differentiation-of-determinant\/","title":{"rendered":"Differentiation of Determinant"},"content":{"rendered":"

Here you will learn differentiation of determinant with example.<\/p>\n

Let’s begin –<\/p>\n

Differentiation of Determinant<\/h2>\n

To differentiate a determinant, we differerentiate one row (or column) at a time, keeping others unchanged.<\/p>\n

for example, if<\/p>\n

D(x) = \\(\\begin{vmatrix} f(x) & g(x) \\\\ u(x) & v(x) \\end{vmatrix}\\) , then<\/p>\n

\\(d\\over dx\\){D(x)} = \\(\\begin{vmatrix} f'(x) & g'(x) \\\\ u(x) & v(x) \\end{vmatrix}\\) + \\(\\begin{vmatrix} f(x) & g(x) \\\\ u'(x) & v'(x) \\end{vmatrix}\\)<\/p>\n

Also,<\/p>\n

\\(d\\over dx\\){D(x)} = \\(\\begin{vmatrix} f'(x) & g(x) \\\\ u'(x) & v(x) \\end{vmatrix}\\) + \\(\\begin{vmatrix} f(x) & g'(x) \\\\ u(x) & v'(x) \\end{vmatrix}\\)<\/p>\n

Similar results holds for the differentiation of determinant of higher order. <\/p>\n

Example<\/span><\/strong> : If f(x) = \\(\\begin{vmatrix} x^2 + a^2 & a \\\\ x^2 + b^2 & b \\end{vmatrix}\\), find f'(x).<\/p>\n

Solution<\/span><\/strong> : We have,<\/p>\n

f(x) = \\(\\begin{vmatrix} x^2 + a^2 & a \\\\ x^2 + b^2 & b \\end{vmatrix}\\)<\/p>\n

\\(\\implies\\) f'(x) = \\(\\begin{vmatrix} 2x & 0 \\\\ x^2 + b^2 & b \\end{vmatrix}\\) + \\(\\begin{vmatrix} x^2 + a^2 & a \\\\ 2x & 0 \\end{vmatrix}\\)<\/p>\n

\\(\\implies\\) f'(x) = {2bx} + {2ax}<\/p>\n

f'(x) = 2x(a +b)<\/p>\n

Example<\/span><\/strong> : If f(x) = \\(\\begin{vmatrix} x + a^2 & ab & ac \\\\ ab & x + b^2 & bc \\\\ ac & bc & x + c^2 \\end{vmatrix}\\), find f'(x).<\/p>\n

Solution<\/span><\/strong> : We have,<\/p>\n

f(x) = \\(\\begin{vmatrix} x + a^2 & ab & ac \\\\ ab & x + b^2 & bc \\\\ ac & bc & x + c^2 \\end{vmatrix}\\)<\/p>\n

\\(\\implies\\) f'(x) = \\(\\begin{vmatrix} 1 & 0 & 0 \\\\ ab & x + b^2 & bc \\\\ ac & bc & x + c^2 \\end{vmatrix}\\) + \\(\\begin{vmatrix} x + a^2 & ab & ac \\\\ 0 & 1 & 0 \\\\ ac & bc & x + c^2 \\end{vmatrix}\\) + \\(\\begin{vmatrix} x + a^2 & ab & ac \\\\ ab & x + b^2 & bc \\\\ 0 & 0 & 1 \\end{vmatrix}\\)<\/p>\n

\\(\\implies\\) f'(x) = \\(\\begin{vmatrix}  x + b^2 & bc \\\\ bc & x + c^2 \\end{vmatrix}\\) + \\(\\begin{vmatrix} x + a^2 & ac \\\\ ac & x + c^2 \\end{vmatrix}\\) + \\(\\begin{vmatrix} x + a^2 & ab \\\\ ab & x + b^2 \\end{vmatrix}\\)<\/p>\n

\\(\\implies\\) f'(x) = {\\((x+b^2)(x+c^2) – b^2c^2\\)} + {\\((x+a^2)(x+c^2) – a^2c^2\\)} + {\\((x+a^2)(x+b^2) – a^2b^2\\)}<\/p>\n

\\(\\implies\\) f'(x) = \\(x^2\\) + x\\(b^2+c^2\\) + \\(x^2\\) + x\\(a^2+c^2\\) + \\(x^2\\) + x\\(a^2+b^2\\)<\/p>\n

f'(x) = \\(3x^2\\) + 2x\\(a^2 + b^2 + c^2\\).<\/p>\n\n\n

\n
Previous – Differentiation of Parametric Functions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn differentiation of determinant with example. Let’s begin – Differentiation of Determinant To differentiate a determinant, we differerentiate one row (or column) at a time, keeping others unchanged. for example, if D(x) = \\(\\begin{vmatrix} f(x) & g(x) \\\\ u(x) & v(x) \\end{vmatrix}\\) , then \\(d\\over dx\\){D(x)} = \\(\\begin{vmatrix} f'(x) & g'(x) \\\\ …<\/p>\n

Differentiation of Determinant<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[36],"tags":[313,275,311,312],"yoast_head":"\nDifferentiation of Determinant - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the differentiation of determinant with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/differentiation-of-determinant\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differentiation of Determinant - 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