{"id":9944,"date":"2022-02-03T18:51:54","date_gmt":"2022-02-03T13:21:54","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9944"},"modified":"2022-02-03T18:51:57","modified_gmt":"2022-02-03T13:21:57","slug":"differentiation-of-constant-proof-and-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differentiation-of-constant-proof-and-examples\/","title":{"rendered":"Differentiation of Constant – Proof and Examples"},"content":{"rendered":"

Here you will learn the differentiation of constant function proof and examples.<\/p>\n

Let’s begin –<\/p>\n

Differentiation of Constant<\/h2>\n

The differentiation of constant function is zero. i.e. \\(d\\over dx\\)(c) = 0.<\/p><\/blockquote>\n

Proof<\/strong> : Let f(x) = c, be a constant function. Then,<\/p>\n

By using first principle,<\/p>\n

\\(d\\over dx\\) (f(x)) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(f(x + h) – f(x)\\over h\\)<\/p>\n

= \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(c – c\\over h\\) = 0<\/p>\n

Hence, \\(d\\over dx\\)(c) = 0, where c is a constant.<\/p>\n

Remark<\/strong> : Geometrically, graph of a constant function is a straight line parallel to x-axis. So, tangent at every point is parallel to x-axis. Consequently slope of the tangent is zero, i.e. \\(dy\\over dx\\) = 0.<\/p>\n

Also Read : Differentiation Formulas Class 12<\/a><\/p>\n

Note<\/strong> : Let f(x) be a differentiable function and let c be a constant. Then c.f(x) is also differentiable such that<\/p>\n

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

i.e. the derivative of a constant times a function is the constant times the derivatives of the function.<\/p>\n

Example<\/strong><\/span> : Differentiate the following with respect to x.<\/p>\n

(i) 5<\/p>\n

(ii) 5x<\/p>\n

(iii) \\(log_x x\\)<\/p>\n

Solution<\/strong><\/span> :<\/p>\n

(i) we have,<\/p>\n

f(x) = 5, which is a constant.<\/p>\n

Therefore, \\(d\\over dx\\)(f(x)) = \\(d\\over dx\\) (5) = 0<\/p>\n

(ii) we have,<\/p>\n

f(x) = 5x<\/p>\n

Therefore, \\(d\\over dx\\)(f(x)) = \\(d\\over dx\\) (5x) = 5\\(d\\over dx\\) (5) = 5(1) = 5<\/p>\n

(iii) we have,<\/p>\n

f(x) = \\(log_x x\\) = 1<\/p>\n

Therefore, \\(d\\over dx\\)(f(x)) = \\(d\\over dx\\) (1) = 0<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn the differentiation of constant function proof and examples. Let’s begin – Differentiation of Constant The differentiation of constant function is zero. i.e. \\(d\\over dx\\)(c) = 0. Proof : Let f(x) = c, be a constant function. Then, By using first principle, \\(d\\over dx\\) (f(x)) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(f(x + h) …<\/p>\n

Differentiation of Constant – Proof and Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[36],"tags":[],"yoast_head":"\nDifferentiation of Constant - Proof and Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn why is the differentiation of constant function is zero with proof and 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-constant-proof-and-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differentiation of Constant - 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