{"id":5133,"date":"2021-09-08T20:38:28","date_gmt":"2021-09-08T15:08:28","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5133"},"modified":"2022-01-16T17:03:18","modified_gmt":"2022-01-16T11:33:18","slug":"logarithmic-differentiation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/logarithmic-differentiation\/","title":{"rendered":"Logarithmic Differentiation – Examples and Formula"},"content":{"rendered":"

Here you will learn formula of logarithmic differentiation with examples.<\/p>\n

Let’s begin –<\/p>\n

Logarithmic Differentiation<\/h2>\n

We have learnt about the derivatives of the functions of the form \\([f(x)]^n\\) , \\(n^{f(x))}\\) and \\(n^n\\) , where f(x) is a function of x and n is a constant. In this section, we will be mainly discussing derivatives of the functions of the form \\([f(x)]^{g(x)}\\) where f(x) and g(x) are functions of x x. To find the derivative of this type of functions we proceed as follows :<\/p>\n

Let y = \\([f(x)]^{g(x)}\\). Taking logarithm of both the sides, we get <\/p>\n

log y = g(x) . log{f(x)}<\/p>\n

Differrentiating with respect to x, we get<\/p>\n

\\(1\\over y\\) \\(dy\\over dx\\) = g(x) \\(\\times\\) \\(1\\over f(x)\\) \\(d\\over dx\\) ((f(x)) + log {f(x)}.\\(d\\over dx\\)(g(x))<\/p>\n

\\(\\therefore\\)  \\(dy\\over dx\\) = y{\\({g(x)\\over f(x)}\\).\\(d\\over dx\\)(f(x)) + log{f(x)}.\\(d\\over dx\\) (g(x))}<\/p>\n

Alternatively, we may write<\/p>\n

y = \\([f(x)]^{g(x)}\\) = \\(e^{g(x)log{f(x)}}\\)<\/p>\n

Differentiating with respect to x, we get<\/p>\n

\\(dy\\over dx\\) = \\(e^{g(x)log{f(x)}}\\) { g(x) \\(\\times\\) \\(1\\over f(x)\\) \\(d\\over dx\\) ((f(x)) + log {f(x)}.\\(d\\over dx\\)(g(x)) }<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\([f(x)]^{g(x)}\\){\\({g(x)\\over f(x)}\\).\\(d\\over dx\\)(f(x)) + log{f(x)}.\\(d\\over dx\\) (g(x))}<\/p>\n

Example<\/span><\/strong> : Differentiate \\(x^x\\) with respect to x.<\/p>\n

Solution<\/span><\/strong> : Let y = \\(x^x\\). Then,<\/p>\n

Taking log both sides,<\/p>\n

log y = x.log x<\/p>\n

\\(\\implies\\) y = \\(e^{x.log x}\\)<\/p>\n

On differentiating both sides with respect to x, we get<\/p>\n

\\(dy\\over dx\\) = \\(e^{x.log x}\\)\\(d\\over dx\\)(xlogx)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(x^x{log x \\times {d\\over dx}(x) + x \\times {d\\over dx}(log x)}\\)<\/p>\n

= \\(x^x(log x + x\\times {1\\over x})\\)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(x^x(1 + logx)\\)<\/p>\n

Example<\/span><\/strong> : Differentiate \\(x^{sinx}\\) with respect to x.<\/p>\n

Solution<\/span><\/strong> : Let y = \\(x^{sinx}\\). Then,<\/p>\n

Taking log both sides,<\/p>\n

log y = sin x.log x<\/p>\n

\\(\\implies\\) y = \\(e^{sin x.log x}\\)<\/p>\n

On differentiating both sides with respect to x, we get<\/p>\n

\\(dy\\over dx\\) = \\(e^{sin x.log x}\\)\\(d\\over dx\\)(sin x.log x)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(x^{sin x}{log x {d\\over dx}(sin x) + sin x {d\\over dx}(log x)}\\)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(x^{sin x}(cos x.log x + {sin x\\over x}\\))<\/p>\n\n\n

\n
Next – Differentiation of Infinite Series Class 12<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Differentiation of Implicit Function<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula of logarithmic differentiation with examples. Let’s begin – Logarithmic Differentiation We have learnt about the derivatives of the functions of the form \\([f(x)]^n\\) , \\(n^{f(x))}\\) and \\(n^n\\) , where f(x) is a function of x and n is a constant. In this section, we will be mainly discussing derivatives of …<\/p>\n

Logarithmic Differentiation – Examples and Formula<\/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":[275,319,320],"yoast_head":"\nLogarithmic Differentiation - Examples and Formula<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula of logarithmic differentiation 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\/logarithmic-differentiation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Logarithmic Differentiation - 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