{"id":5123,"date":"2021-09-08T18:18:43","date_gmt":"2021-09-08T12:48:43","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5123"},"modified":"2021-11-22T22:09:11","modified_gmt":"2021-11-22T16:39:11","slug":"product-rule-in-differentiation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/product-rule-in-differentiation\/","title":{"rendered":"Product Rule in Differentiation with Examples"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Product Rule in Differentiation<\/h2>\n

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

\n

\\(d\\over dx\\) {f(x) g(x)} = \\(d\\over dx\\) (f(x)) g(x) + f(x). \\(d\\over dx\\) (g(x))<\/strong><\/p>\n<\/blockquote>\n

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

\n

\\(d\\over dx\\) (f(x) g(x) h(x)) = \\(d\\over dx\\) (f(x)) g(x) h(x) + f(x). \\(d\\over dx\\) (g(x)) h(x) + f(x) g(x) \\(d\\over dx\\) (h(x)) <\/strong> <\/p>\n<\/blockquote>\n

Example 1<\/strong><\/span> : find the differentiation of sinx cosx.<\/p>\n

Solution<\/strong> <\/span>: Let sinx = f(x) and g(x) = cosx<\/p>\n

Then, by using product rule in differentiation,<\/p>\n

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

\\(d\\over dx\\) [sinx.cosx] = \\(d\\over dx\\) (sinx) cosx + sinx. \\(d\\over dx\\) (cosx)<\/p>\n

= cosx cosx + sinx (-sinx)<\/p>\n

= \\(cos^2x\\) – \\(sin^2x\\)<\/p>\n

= cos2x<\/p>\n

Example 2<\/strong><\/span> : find the differentiation of x sinx.<\/p>\n

Solution<\/strong> <\/span>: Let x = f(x) and g(x) = sinx<\/p>\n

Then, by using product rule in differentiation,<\/p>\n

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

\\(d\\over dx\\) [x.sinx] = \\(d\\over dx\\) (x) sinx + x. \\(d\\over dx\\) (sinx)<\/p>\n

= 1.sinx + x.(cosx)<\/p>\n

= sinx + x cosx<\/p>\n

Example 3<\/strong><\/span> : find the differentiation of \\(e^x log\\sqrt{x} tanx\\).<\/p>\n

Solution<\/strong> <\/span>: Let \\(e^x\\) = f(x) , g(x) = \\(log\\sqrt{x}\\) and h(x) = tanx<\/p>\n

Then, by using product rule,<\/p>\n

\\(d\\over dx\\) {f(x) g(x) h(x)} = \\(d\\over dx\\) (f(x)) g(x) h(x) + f(x). \\(d\\over dx\\) (g(x)) h(x) + f(x) g(x) \\(d\\over dx\\) (h(x))<\/p>\n

\\(d\\over dx\\) [ \\(e^x log\\sqrt{x} tanx\\)] = \\(d\\over dx\\) [ \\(e^x \\times {1\\over 2} logx \\times tanx\\)]<\/p>\n

= \\(1\\over 2\\) \\(d\\over dx\\) [ \\(e^x logx  tanx\\)]<\/p>\n

= \\(1\\over 2\\) [{\\(d\\over dx\\) (\\(e^x\\))} logx tanx  + \\(e^x\\). {\\(d\\over dx\\) (logx)} tanx + \\(e^x\\) logx {\\(d\\over dx\\) (tanx)}]<\/p>\n

= \\(1\\over 2\\) { \\(e^x\\) logx tanx  + \\(e^x\\). {\\(1\\over x\\)} tanx + \\(e^x\\) logx \\(sec^2x\\)}<\/p>\n

= \\(1\\over 2\\) \\(e^x\\) { logx tanx  + \\(tanx\\over x\\) + logx \\(sec^2x\\)}<\/p>\n\n\n

\n
Next – Quotient Rule in Differentiation with Examples<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Differentiation Formulas Class 12<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is product rule in differentiation with examples. Let’s begin – Product Rule in Differentiation If f(x) and g(x) are differentiable functions, then f(x)g(x) is also differentiable function such that \\(d\\over dx\\) {f(x) g(x)} = \\(d\\over dx\\) (f(x)) g(x) + f(x). \\(d\\over dx\\) (g(x)) If f(x), g(x) and h(x) are differentiable …<\/p>\n

Product 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":[275,328,330,329],"yoast_head":"\nProduct Rule in Differentiation with Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is product 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\/product-rule-in-differentiation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Product Rule in Differentiation with Examples - 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