{"id":5586,"date":"2019-10-21T15:20:50","date_gmt":"2019-10-21T09:50:50","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=5586"},"modified":"2022-11-04T12:06:29","modified_gmt":"2022-11-04T06:36:29","slug":"differentiation-formulas","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/differentiation-formulas\/","title":{"rendered":"Differentiation formulas |Derivatives of Function list"},"content":{"rendered":"\n
\"Differentiation<\/figure><\/div>\n\n\n\n

Here is the list of Differentiation formulas|Derivatives of Function to remember to score well in your Mathematics examination. The formula list include the derivative of polynomial functions, trigonometric functions,inverse trigonometric function, Logarithm function,exponential function. This also includes the rules for finding the derivative of various composite function and difficult function,<\/p>\n\n\n\n

Differentiation from First Principle<\/h2>\n\n\n\n

$\\frac {d}{dx} f(x) =\\lim_{h\\rightarrow 0} \\frac{f(x+h) -f(x)}{h}$<\/p>\n\n\n\n

Standard Differentiation formulas<\/h2>\n\n\n\n

$\\frac {d}{dx} (c) = 0$  ( Where c is a constant)<\/p>\n\n\n\n

$\\frac {d}{dx} (cx) = c$ ( Where c is a constant)<\/p>\n\n\n\n

$\\frac {d}{dx} (x^n) = nx^{n-1}$<\/p>\n\n\n\n

$\\frac {d}{dx} (e^x) = e^x$<\/p>\n\n\n\n

$\\frac {d}{dx} (ln x) = \\frac {1}{x}$<\/p>\n\n\n\n

$\\frac {d}{dx} (log_{10} x) =\\frac {1}{x ln 10}$<\/p>\n\n\n\n

$\\frac {d}{dx} (log_{a} x) =\\frac {1}{x ln a}$<\/p>\n\n\n\n

$\\frac {d}{dx} (a^x) = a^x ln a$<\/p>\n\n\n\n

Differentiation formulas for Trigonometric Functions<\/h2>\n\n\n\n

$\\frac {d}{dx} (sinx) = cos x$<\/p>\n\n\n\n

$\\frac {d}{dx} (cos x) = -sin x$<\/p>\n\n\n\n

$\\frac {d}{dx} (tan x) = sec^2 x, x \\ne (2n +1) \\frac {\\pi}{2}, n \\in I$<\/p>\n\n\n\n

$\\frac {d}{dx} (cot x) = -cosec^2 x, x \\ne n \\pi, n \\in I$<\/p>\n\n\n\n

$\\frac {d}{dx} (sec x) = sec(x) tan(x) ,x \\ne (2n +1) \\frac {\\pi}{2}, n \\in I$<\/p>\n\n\n\n

$\\frac {d}{dx} (cosec x) = -cosec(x) cot(x) , x \\ne n \\pi, n \\in I$<\/p>\n\n\n\n

Differentiation formulas for Inverse Trigonometric Functions<\/h2>\n\n\n\n

$\\frac {d}{dx} (sin^{-1}x) = \\frac {1}{\\sqrt {1-x^2} }, -1< x< 1$<\/p>\n\n\n\n

$\\frac {d}{dx} (cos ^{-1}x) = -\\frac {1}{\\sqrt {1-x^2}} , -1< x< 1$<\/p>\n\n\n\n

$\\frac {d}{dx} (tan ^{-1}x) = \\frac {1}{1 + x^2}$<\/p>\n\n\n\n

$\\frac {d}{dx} (cot ^{-1}x) = -\\frac {1}{1 + x^2}$<\/p>\n\n\n\n

$\\frac {d}{dx} (sec ^{-1}x) = \\frac {1}{|x|\\sqrt {x^-1}} , |x| > 1$<\/p>\n\n\n\n

$\\frac {d}{dx} (cosec ^{-1}x) = \\frac {1}{|x|\\sqrt {x^-1}} , |x| > 1$<\/p>\n\n\n\n

Algebra of derivatives<\/h2>\n\n\n\n

Multiplication by Constant<\/p>\n\n\n\n

$\\frac {d}{dx} [cf(x)] = c \\frac {d}{dx} f(x) $<\/p>\n\n\n\n

Example<\/em><\/p>\n\n\n\n

$\\frac {d}{dx} [2 sinx ] = 2 \\frac {d}{dx} sin x =2 cos (x)$<\/p>\n\n\n\n

Addition and Subtraction<\/em><\/p>\n\n\n\n

$\\frac {d}{dx} [f(x)+g(x)]=\\frac {d}{dx} f(x) + \\frac {d}{dx} g(x)$<\/p>\n\n\n\n

$\\frac {d}{dx} [f(x)-g(x)]=\\frac {d}{dx} f(x) – \\frac {d}{dx} g(x)$<\/p>\n\n\n\n

Example<\/em><\/p>\n\n\n\n

$\\frac {d}{dx} [sinx -cos x ] =  \\frac {d}{dx} sin x – \\frac {d}{dx} cos x =cos (x) + sin(x)$<\/p>\n\n\n\n

Multiplication<\/em><\/p>\n\n\n\n

$\\frac {d}{dx} [f(x)g(x)]=g(x) \\frac {d}{dx} f(x) + f(x) \\frac {d}{dx} g(x)$<\/p>\n\n\n\n

Example<\/em><\/p>\n\n\n\n

$\\frac {d}{dx} [x^2 sinx  ] = x^2 \\frac {d}{dx} sin x  + sin x \\frac {d}{dx} x^2 =x^2 cos (x) + 2x sin(x)$<\/p>\n\n\n\n

Division <\/em><\/p>\n\n\n\n

$\\frac {d}{dx} [f(x)\/g(x)]=\\frac {g(x) \\frac {d}{dx} f(x) – f(x) \\frac {d}{dx} g(x)}{[g(x)]^2} $<\/p>\n\n\n\n

Example<\/p>\n\n\n\n

$\\frac {d}{dx} [sin(x) \/x^2]=\\frac {x^2 \\frac {d}{dx} sin(x)  – sin(x) \\frac {d}{dx} x^2}{x^4} $<\/p>\n\n\n\n

$=\\frac {x^2 cos (x) – 2x sin(x)}{x^4}$<\/p>\n\n\n\n

$=\\frac {x cos (x) – 2 sin(x)}{x^3}$<\/p>\n\n\n\n

Chain Rule<\/h2>\n\n\n\n

if y = f(u) and u =g(x) ,then<\/p>\n\n\n\n

$\\frac {dy}{dx} = \\frac {dy}{du} \\frac {du}{dx}$<\/p>\n\n\n\n

Example<\/p>\n\n\n\n

$\\frac {d}{dx} [sin (x^3)] = \\frac {d}{du} sin (u)  \\frac {d}{dx} (x^3)= 3x^2 cos (x^3)$<\/p>\n\n\n\n

Differentiable at a point a<\/h2>\n\n\n\n

$\\lim_{h \\rightarrow 0 -0} \\frac{f(a+h) -f(a)}{h} = \\lim_{h \\rightarrow 0 +0} \\frac{f(a+h) -f(a)}{h}= \\text {finite number}$<\/p>\n\n\n\n

Logarithmic Differentiation<\/h2>\n\n\n\n

If  $ y = \\frac {f_1(x) f_2(x) f_3(x)}{ g_1(x) g_2(x) g_3(x)}$<\/p>\n\n\n\n

Then we first take logarithm and then differentiate it<\/p>\n\n\n\n

nth Derivatives of Common Function<\/h2>\n\n\n\n

$\\frac {d^n}{dx^n} [sin(ax + b)]= a^n sin (\\frac {n\\pi}{2} + ax + b)$<\/p>\n\n\n\n

$\\frac {d^n}{dx^n} [cos(ax + b)]= a^n cos (\\frac {n\\pi}{2} + ax + b)$<\/p>\n\n\n\n

$\\frac {d^n}{dx^n} [e^{ax}]= a^n e^{ax}$<\/p>\n\n\n\n

Also Reads<\/strong>
Trigonometry Formulas for class 11 (PDF download)<\/a>
sin cos tan table<\/a>
Integration Formulas<\/a>
sin 18 degrees<\/a>
tan 15 degrees<\/a>
Inverse Trigonometric Function Formulas<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Here is the list of Differentiation formulas|Derivatives of Function to remember to score well in your Mathematics examination. The formula list include the derivative of polynomial functions, trigonometric functions,inverse trigonometric function, Logarithm function,exponential function. This also includes the rules for finding the derivative of various composite function and difficult function, Differentiation from First Principle $\\frac […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[14],"tags":[],"class_list":["post-5586","post","type-post","status-publish","format-standard","hentry","category-physics"],"yoast_head":"\nDifferentiation formulas |Derivatives of Function list - 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The formula list include the derivative of polynomial functions, trigonometric functions,inverse trigonometric function, Logarithm function,exponential function. This also includes the rules for finding the derivative of various composite function and difficult function, Differentiation from First Principle $\\frac…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5586","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=5586"}],"version-history":[{"count":4,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5586\/revisions"}],"predecessor-version":[{"id":8303,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5586\/revisions\/8303"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=5586"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=5586"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=5586"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}