{"id":5125,"date":"2021-09-08T18:20:32","date_gmt":"2021-09-08T12:50:32","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5125"},"modified":"2021-11-22T22:06:48","modified_gmt":"2021-11-22T16:36:48","slug":"quotient-rule-in-differentiation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/quotient-rule-in-differentiation\/","title":{"rendered":"Quotient Rule in Differentiation with Examples"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Quotient Rule in Differentiation<\/h2>\n

If f(x) and g(x) are two differentiable functions and g(x) \\(\\ne\\) 0, then<\/p>\n

\n

\\(d\\over dx\\) {\\(f(x)\\over g(x)\\)} = \\({g(x) {d\\over dx} (f(x)) – f(x) {d\\over dx} (g(x))}\\over {(g(x))^2}\\)<\/p>\n<\/blockquote>\n

Example 1<\/span><\/strong> : find the differentiation of \\(sinx\\over {x + 1}\\).<\/p>\n

Solution <\/span><\/strong>: Let f(x)  = sinx and g(x) = x + 1<\/p>\n

By using quotient rule in differentiation,<\/p>\n

\\(d\\over dx\\) {\\(f(x)\\over g(x)\\)} = \\({g(x) {d\\over dx} (f(x)) – f(x) {d\\over dx} (g(x))}\\over {(g(x))^2}\\)<\/p>\n

\\(d\\over dx\\) {\\(sinx\\over x + 1\\)} = \\({(x + 1) {d\\over dx} (sinx) – sinx {d\\over dx} (x + 1)}\\over {(x + 1)^2}\\)<\/p>\n

= \\({(x + 1)(cosx) – sinx.1}\\over {(x + 1)^2}\\)<\/p>\n

= \\((x+1)cosx – sinx\\over {(x + 1)^2}\\)<\/p>\n

Example 2<\/span><\/strong> : find the differentiation of \\(e^x + sinx\\over 1 + logx\\).<\/p>\n

Solution <\/span><\/strong>: Let f(x)  = \\(e^x + sinx\\) and g(x) = 1 + logx<\/p>\n

By using quotient rule,<\/p>\n

\\(d\\over dx\\) {\\(f(x)\\over g(x)\\)} = \\({g(x) {d\\over dx} (f(x)) – f(x) {d\\over dx} (g(x))}\\over {(g(x))^2}\\)<\/p>\n

\\(d\\over dx\\) {\\(e^x + sinx\\over 1 + logx\\)} = \\({(1 + logx) {d\\over dx} (e^x + sinx) – (e^x + sinx) {d\\over dx} (1 + logx)}\\over {(1 + logx)^2}\\)<\/p>\n

= \\({(1 + logx) (e^x + cosx) – (e^x + sinx) (0 + {1\\over x})}\\over {(1 + logx)^2}\\)<\/p>\n

= \\({(1 + logx) (e^x + cosx) – (e^x + {sinx\\over x})}\\over {(1 + logx)^2}\\)<\/p>\n

Example 3<\/span><\/strong> : find the differentiation of \\(sinx – xcosx\\over {xsinx + cosx}\\).<\/p>\n

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

By using quotient rule,<\/p>\n

\\(d\\over dx\\) {\\(f(x)\\over g(x)\\)} = \\({g(x) {d\\over dx} (f(x)) – f(x) {d\\over dx} (g(x))}\\over {(g(x))^2}\\)<\/p>\n

\\(d\\over dx\\) {\\(sinx – xcosx\\over {xsinx + cosx}\\)}<\/p>\n

= \\({(xsinx + cosx) {d\\over dx} (sinx – xcosx) – (sinx – xcosx) {d\\over dx} (xsinx + cosx)}\\over {(xsinx + cosx)^2}\\)<\/p>\n

= \\({(xsinx + cosx) (cosx – cosx + xsinx) – (sinx – xcosx) (sinx + xcosx – sinx)}\\over {(xsinx + cosx)^2}\\)<\/p>\n

= \\({(xsinx + cosx) (xsinx) – (sinx – xcosx) (xcosx)}\\over {(xsinx + cosx)^2}\\)<\/p>\n

= \\((x^2sin^2x + xsinx cosx) – (xsinx cosx – x^2cos^2x)\\over {(xsinx + cosx)^2}\\)<\/p>\n

= \\(x^2(sin^2x + cos^2x)\\over {(xsinx + cosx)^2}\\)<\/p>\n

=\\(x^2\\over {(xsinx + cosx)^2}\\)<\/p>\n\n\n

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

Here you will learn what is quotient rule in differentiation with examples. Let’s begin – Quotient Rule in Differentiation If f(x) and g(x) are two differentiable functions and g(x) \\(\\ne\\) 0, then \\(d\\over dx\\) {\\(f(x)\\over g(x)\\)} = \\({g(x) {d\\over dx} (f(x)) – f(x) {d\\over dx} (g(x))}\\over {(g(x))^2}\\) Example 1 : find the differentiation of \\(sinx\\over …<\/p>\n

Quotient 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":[275,328,327],"yoast_head":"\nQuotient Rule in Differentiation with Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is quotient 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\/quotient-rule-in-differentiation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Quotient Rule in Differentiation with Examples - 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