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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} \log_a u = \frac{log_a e}{u} \frac{du}{dx} \qquad a \neq 0,1</math> |
+ | |- | ||
+ | | <math> \frac{d}{dx} \ln u = \frac{d}{dx} log_e u = \frac{1}{u} \frac{du}{dx}</math> | ||
+ | |- | ||
+ | | <math> \frac{d}{dx} a^u = a^u \ln a \frac{du}{dx} </math> | ||
+ | |- | ||
+ | | <math> \frac{d}{dx} e^u = e^u \frac{du}{dx} </math> | ||
+ | |- | ||
| <span class="texhtml">''e''<sup>''u''</sup></span> | | <span class="texhtml">''e''<sup>''u''</sup></span> | ||
| <math>e^u \frac{du}{dx}</math> | | <math>e^u \frac{du}{dx}</math> |
Revision as of 21:35, 11 November 2010
Table of Derivatives | |
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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 $ |
Please continue | write a rule here |
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} $ credit |
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 | ||
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$ \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 ) $ | ||
add function here | derivative here | |
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} $ | ||
eu | $ e^u \frac{du}{dx} $ | |
add function here | derivative here | |
Derivatives of hyperbolic functions | ||
hyperbolic sine | $ \text{sh } u $ | $ \text{ch } u \frac{du}{dx} $ |
add function here | derivative here |