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| + | [[Category:Formulas]] | ||
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| + | keywords:quotient rule, chain rule, Leibniz rule | ||
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| + | <center><font size= 4> | ||
| + | '''[[Collective_Table_of_Formulas|Collective Table of Formulas]]''' | ||
| + | </font size> | ||
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| + | '''Derivatives''' | ||
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| + | click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]] | ||
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| + | </center> | ||
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| + | ---- | ||
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{| | {| | ||
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | General Rules | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | General Rules | ||
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| <math>\frac{d}{dx}\left( c_1 u_1+c_2 u_2 \right) = c_1 \frac{d}{dx}\left( u_1 \right)+c_2 \frac{d}{dx}\left( u_2 \right), \ \text{ for any constants }c_1, c_2</math> | | <math>\frac{d}{dx}\left( c_1 u_1+c_2 u_2 \right) = c_1 \frac{d}{dx}\left( u_1 \right)+c_2 \frac{d}{dx}\left( u_2 \right), \ \text{ for any constants }c_1, c_2</math> | ||
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| − | | | + | | Quotient rule |
| − | | | + | |<math> \frac{d}{dx} ( \frac{u}{v} ) = \frac{v ( \frac{du}{dx} ) - u ( \frac{dv}{dx} )}{v^2}</math> |
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| + | | Exponent rule | ||
| + | |<math> \frac{d}{dx} ( u^n ) = n u^{n-1} \frac{du}{dx}</math> | ||
| + | |- | ||
| + | | Chain rule | ||
| + | |<math> \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}</math> | ||
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| + | |<math> \frac{du}{dx} = \frac{1}{\frac{dx}{du}}</math> | ||
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| + | |<math> \frac{dy}{dx} = \frac{dy}{du}/\frac{dx}{du}</math> | ||
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Leibnitz Rule for Successive Derivatives of a Product | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Leibnitz Rule for Successive Derivatives of a Product | ||
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| third order | | third order | ||
| − | | <math>\frac{d^3}{dx^3}\left( u v \right)= u \frac{d^3v }{dx^3} + 3 \frac{du }{dx}\frac{d^2v }{dx^2}+ 3 \frac{du^2 }{dx^2}\frac{d v }{dx}+ v \frac{d^3u }{dx^3} </math> | + | | <math>\frac{d^3}{dx^3}\left( u v \right)= u \frac{d^3v }{dx^3} + 3 \frac{du }{dx}\frac{d^2v }{dx^2}+ 3 \frac{du^2 }{dx^2}\frac{d v }{dx}+ v \frac{d^3u }{dx^3} </math> |
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| n-th order | | n-th order | ||
| <math>\frac{d^n}{dx^n}\left( u v \right)= u \frac{d^n v }{dx^n} + \left( \begin{array}{cc}n \\ 1 \end{array}\right) \frac{du }{dx}\frac{d^{n-1}v }{dx^{n-1}} + \left( \begin{array}{cc}n \\ 2 \end{array}\right) \frac{d^2u}{dx^2}\frac{d^{n-2}v }{dx^{n-2}}+ \ldots + v \frac{d^n u }{dx^n}</math> | | <math>\frac{d^n}{dx^n}\left( u v \right)= u \frac{d^n v }{dx^n} + \left( \begin{array}{cc}n \\ 1 \end{array}\right) \frac{du }{dx}\frac{d^{n-1}v }{dx^{n-1}} + \left( \begin{array}{cc}n \\ 2 \end{array}\right) \frac{d^2u}{dx^2}\frac{d^{n-2}v }{dx^{n-2}}+ \ldots + v \frac{d^n u }{dx^n}</math> | ||
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| <math> \frac {d}{dx} \arccot u = - \frac{1}{1+u^2} \frac{du}{dx} \qquad ( 0 < \arccot u < \pi ) </math> | | <math> \frac {d}{dx} \arccot u = - \frac{1}{1+u^2} \frac{du}{dx} \qquad ( 0 < \arccot u < \pi ) </math> | ||
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="3" | Derivatives of exponential and logarithm functions | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="3" | Derivatives of exponential and logarithm functions | ||
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| <math> \frac{d}{dx} u^v = \frac{d}{dx} e^{v ln u} = e^{v ln u} \frac {d}{dx} [ v ln u ] = v u^{v-1} \frac{du}{dx} + u^v ln u \frac{dv}{dx}</math> | | <math> \frac{d}{dx} u^v = \frac{d}{dx} e^{v ln u} = e^{v ln u} \frac {d}{dx} [ v ln u ] = v u^{v-1} \frac{du}{dx} + u^v ln u \frac{dv}{dx}</math> | ||
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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="3" | Derivatives of hyperbolic functions | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="3" | Derivatives of hyperbolic functions | ||
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| − | | <math> \frac{d}{dx} \ | + | | <math> \frac{d}{dx} \sinh u = \cosh u \frac{du}{dx}</math> |
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| − | | <math> \frac{d}{dx} \ | + | | <math> \frac{d}{dx} \cosh u = \sinh u \frac{du}{dx}</math> |
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| − | | <math> \frac{d}{dx} \ | + | | <math> \frac{d}{dx} \tanh u = \frac{1}{\cosh^2 u} \frac{du}{dx}</math> |
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| − | | <math> \frac{d}{dx} \ | + | | <math> \frac{d}{dx} \coth u = - \frac{1}{\sinh^2 u} \frac{du}{dx}</math> |
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| − | | <math> \frac{d}{dx} \frac{1}{\ | + | | <math> \frac{d}{dx} \frac{1}{\cosh u} = - \frac{\tanh u}{\cosh u} \frac{du}{dx} </math> |
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| − | | <math> \frac{d}{dx} \frac{1}{\ | + | | <math> \frac{d}{dx} \frac{1}{\sinh u} = - \frac{\coth u}{\sinh u} \frac{du}{dx} </math> |
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| − | | <math> \frac{d}{dx} \ | + | | <math> \frac{d}{dx}\ \operatorname{arsinh}\ u = \frac{1}{\sqrt{u^2+1}} \frac{du}{dx}</math> |
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| − | | <math> \frac{d}{dx} \ | + | | <math> \frac{d}{dx}\ \operatorname{arcosh}\ u = \frac{1}{\sqrt{u^2-1}} \frac{du}{dx}</math> |
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| − | | <math> \frac{d}{dx} \ | + | | <math> \frac{d}{dx}\ \operatorname{artanh}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ -1 < u < 1 \ ) </math> |
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| − | | <math> \frac{d}{dx} \ | + | | <math> \frac{d}{dx}\ \operatorname{arcoth}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ u > 1 \ or \ u < -1 \ ) </math> |
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[[Collective_Table_of_Formulas|Back to Collective Table of Formulas]] | [[Collective_Table_of_Formulas|Back to Collective Table of Formulas]] | ||
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Latest revision as of 16:40, 27 February 2015
keywords:quotient rule, chain rule, Leibniz rule
Derivatives
click here for more formulas
| General Rules | |
|---|---|
| Derivative of a constant | $ \frac{d}{dx}\left( c \right) = 0, \ \text{ for any constant }c $ |
| $ \frac{d}{dx}\left( c x \right) = c, \ \text{ for any constant }c $ | |
| Linearity | $ \frac{d}{dx}\left( c_1 u_1+c_2 u_2 \right) = c_1 \frac{d}{dx}\left( u_1 \right)+c_2 \frac{d}{dx}\left( u_2 \right), \ \text{ for any constants }c_1, c_2 $ |
| Quotient rule | $ \frac{d}{dx} ( \frac{u}{v} ) = \frac{v ( \frac{du}{dx} ) - u ( \frac{dv}{dx} )}{v^2} $ |
| Exponent rule | $ \frac{d}{dx} ( u^n ) = n u^{n-1} \frac{du}{dx} $ |
| Chain rule | $ \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} $ |
| $ \frac{du}{dx} = \frac{1}{\frac{dx}{du}} $ | |
| $ \frac{dy}{dx} = \frac{dy}{du}/\frac{dx}{du} $ | |
| Leibnitz Rule for Successive Derivatives of a Product | |
| first order | $ \frac{d}{dx}\left( u v \right)= u \frac{dv }{dx} + v \frac{du }{dx} $ |
| second order | $ \frac{d^2}{dx^2}\left( u v \right)= u \frac{d^2v }{dx^2} + 2\frac{du }{dx}\frac{dv }{dx}+ v \frac{d^2u }{dx^2} $ |
| third order | $ \frac{d^3}{dx^3}\left( u v \right)= u \frac{d^3v }{dx^3} + 3 \frac{du }{dx}\frac{d^2v }{dx^2}+ 3 \frac{du^2 }{dx^2}\frac{d v }{dx}+ v \frac{d^3u }{dx^3} $ |
| n-th order | $ \frac{d^n}{dx^n}\left( u v \right)= u \frac{d^n v }{dx^n} + \left( \begin{array}{cc}n \\ 1 \end{array}\right) \frac{du }{dx}\frac{d^{n-1}v }{dx^{n-1}} + \left( \begin{array}{cc}n \\ 2 \end{array}\right) \frac{d^2u}{dx^2}\frac{d^{n-2}v }{dx^{n-2}}+ \ldots + v \frac{d^n u }{dx^n} $ |
| Derivatives of trigonometric functions | ||
|---|---|---|
| $ \frac {d}{dx} \sin u = \cos u \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \cos u = - \sin u \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \tan u = \frac{1}{\cos^2 u} \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \cot u = - \frac{1}{\sin^2 u} \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \frac{1}{\cos u} = \frac{\tan u}{\cos u} \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \frac{1}{\sin u} = - \frac{\cot u}{\sin u} \frac{du}{dx} $ | ||
| $ \frac {d}{dx} \arcsin u = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} \qquad ( - \frac{\pi}{2} < \arcsin u < \frac{\pi}{2} ) $ | ||
| $ \frac {d}{dx} \arccos u = - \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} \qquad ( 0 < \arccos u < \pi ) $ | ||
| $ \frac {d}{dx} \arctan u = \frac{1}{1+u^2} \frac{du}{dx} \qquad ( - \frac{\pi}{2} < \arctan u < \frac{\pi}{2} ) $ | ||
| $ \frac {d}{dx} \arccot u = - \frac{1}{1+u^2} \frac{du}{dx} \qquad ( 0 < \arccot u < \pi ) $ | ||
| Derivatives of exponential and logarithm functions | ||
| $ \frac{d}{dx} \log_a u = \frac{log_a e}{u} \frac{du}{dx} \qquad a \neq 0,1 $ | ||
| $ \frac{d}{dx} \ln u = \frac{d}{dx} log_e u = \frac{1}{u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx} a^u = a^u \ln a \frac{du}{dx} $ | ||
| $ \frac{d}{dx} e^u = e^u \frac{du}{dx} $ | ||
| $ \frac{d}{dx} u^v = \frac{d}{dx} e^{v ln u} = e^{v ln u} \frac {d}{dx} [ v ln u ] = v u^{v-1} \frac{du}{dx} + u^v ln u \frac{dv}{dx} $ | ||
| Derivatives of hyperbolic functions | ||
| $ \frac{d}{dx} \sinh u = \cosh u \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \cosh u = \sinh u \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \tanh u = \frac{1}{\cosh^2 u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \coth u = - \frac{1}{\sinh^2 u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \frac{1}{\cosh u} = - \frac{\tanh u}{\cosh u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx} \frac{1}{\sinh u} = - \frac{\coth u}{\sinh u} \frac{du}{dx} $ | ||
| $ \frac{d}{dx}\ \operatorname{arsinh}\ u = \frac{1}{\sqrt{u^2+1}} \frac{du}{dx} $ | ||
| $ \frac{d}{dx}\ \operatorname{arcosh}\ u = \frac{1}{\sqrt{u^2-1}} \frac{du}{dx} $ | ||
| $ \frac{d}{dx}\ \operatorname{artanh}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ -1 < u < 1 \ ) $ | ||
| $ \frac{d}{dx}\ \operatorname{arcoth}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ u > 1 \ or \ u < -1 \ ) $ | ||
