{"id":5127,"date":"2021-09-08T18:29:08","date_gmt":"2021-09-08T12:59:08","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5127"},"modified":"2021-11-22T22:07:12","modified_gmt":"2021-11-22T16:37:12","slug":"chain-rule-in-differentiation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/chain-rule-in-differentiation\/","title":{"rendered":"Chain Rule in Differentiation with Examples"},"content":{"rendered":"

Here you will learn what is chain rule in differentiation with examples.<\/p>\n

Let’s begin –<\/p>\n

Chain Rule in Differentiation<\/h2>\n

If f(x) and g(x) are differentiable functions, then fog is also differentiable and<\/p>\n

\n

(fog)'(x) = f'(x) = f'(g(x)) g'(x)<\/p>\n

or, \\(d\\over dx\\) {(fog) (x)} = \\(d\\over d g(x)\\) {(fog) (x)} \\(d\\over dx\\) (g(x)).<\/p>\n<\/blockquote>\n

Remark 1 <\/strong>: The above rule can also be restated as follows :<\/p>\n

\n

If z = f(y) and y = g(x), then \\(dz\\over dx\\) = \\(dz\\over dy\\).\\(dy\\over dx\\)<\/p>\n

Derivative of z with respect to x = (Derivative of z with respect to y) \\(\\times\\) (Derivative of y with respect to x)<\/p>\n<\/blockquote>\n

Remark 2<\/strong> : This chain rule can be extended further.<\/p>\n

\n

Derivative of z with respect to x = (Derivative of z with respect to u) \\(\\times\\) (Derivative of u with respect to v) \\(\\times\\) (Derivatve of v with respect to x)<\/p>\n<\/blockquote>\n

Example 1<\/span><\/strong> : Differentiate \\(sin(x^2 + 1)\\) with respect to x.<\/p>\n

Solution<\/span><\/strong> : Let y = \\(sin(x^2 + 1)\\). Putting u = \\(x^2 + 1\\) , we get<\/p>\n

y = sin u and u = \\(x^2 + 1\\)<\/p>\n

\\(\\therefore\\)  \\(dy\\over du\\) = cosu and \\(du\\over dx\\) = 2x<\/p>\n

Now, \\(dy\\over dx\\) = \\(dy\\over du\\) \\(\\times\\) \\(du\\over dx\\)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = (cos u)2x = 2x \\(cos (x^2 + 1)\\)<\/p>\n

Hence, \\(d\\over dx\\) [\\(sin(x^2+1)\\)] = 2x \\(cos (x^2 + 1)\\)<\/p>\n

Example 2<\/span><\/strong> : Differentiate \\(e^{sinx}\\) with respect to x.<\/p>\n

Solution<\/span><\/strong> : Let y = \\(e^{sinx}\\). Putting u = sinx , we get<\/p>\n

y = \\(e^u\\) and u = sinx<\/p>\n

\\(\\therefore\\)  \\(dy\\over du\\) = \\(e^u\\) and \\(du\\over dx\\) = cosx<\/p>\n

Now, \\(dy\\over dx\\) = \\(dy\\over du\\) \\(\\times\\) \\(du\\over dx\\)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(e^u\\)cosx = \\(e^{sinx}\\)cosx<\/p>\n

Hence, \\(d\\over dx\\) [\\(e^{sinx}\\)] = \\(e^{sinx}\\)cosx<\/p>\n

Example 3<\/span><\/strong> : Differentiate log sinx with respect to x.<\/p>\n

Solution<\/span><\/strong> : Let y = log u. Putting u = sinx , we get<\/p>\n

y = log u and u = sinx<\/p>\n

\\(\\therefore\\)  \\(dy\\over du\\) = \\(1\\over u\\) and \\(du\\over dx\\) = cosx<\/p>\n

Now, \\(dy\\over dx\\) = \\(dy\\over du\\) \\(\\times\\) \\(du\\over dx\\)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(1\\over u\\)cosx = \\(1\\over sinx\\)cosx = cotx<\/p>\n

Hence, \\(d\\over dx\\) [log sinx] = cotx<\/p>\n\n\n

\n
Next – Differentiation of Inverse Trigonometric Functions<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Quotient Rule in Differentiation with Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is chain rule in differentiation with examples. Let’s begin – Chain Rule in Differentiation If f(x) and g(x) are differentiable functions, then fog is also differentiable and (fog)'(x) = f'(x) = f'(g(x)) g'(x) or, \\(d\\over dx\\) {(fog) (x)} = \\(d\\over d g(x)\\) {(fog) (x)} \\(d\\over dx\\) (g(x)). Remark 1 : The …<\/p>\n

Chain Rule in Differentiation with Examples<\/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":[326,325,275,328],"yoast_head":"\nChain Rule in Differentiation with Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is chain rule in 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\/chain-rule-in-differentiation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Chain Rule in Differentiation with Examples - 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