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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 trigonometric 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 trigonometric functions | ||
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| − | | | + | | <math> \frac {d}{dx} \sin u = \cos u \frac{du}{dx} </math> |
| + | |- | ||
| + | | <math> \frac {d}{dx} \cos u = - \sin u \frac{du}{dx} </math> | ||
| + | |- | ||
| + | | <math> \frac {d}{dx} \tan u = \frac{1}{\cos^2 u} \frac{du}{dx} </math> | ||
| + | |- | ||
| + | | <math> \frac {d}{dx} \arcsin u = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} \qquad ( - \frac{\pi}{2} < \arcsin u < \frac{\pi}{2} )</math> | ||
| + | |- | ||
| + | | <math> \frac {d}{dx} \arccos u = - \frac{1}{\sqrt{1-u^2}} \frac{du}{dx} \qquad ( 0 < \arccos u < \pi ) </math> | ||
| + | |- | ||
| + | |||
| + | |||
| <span class="texhtml">sin'' u''</span> | | <span class="texhtml">sin'' u''</span> | ||
| align="left" | <math>\cos u \frac{du}{dx}</math> | | align="left" | <math>\cos u \frac{du}{dx}</math> | ||
Revision as of 20:14, 11 November 2010
| Table of Derivatives | |
|---|---|
| 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 | ||
|---|---|---|
| $ \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} \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 ) $ | ||
| sin u | $ \cos u \frac{du}{dx} $ | |
| add function here | derivative here | |
| Derivatives of exponential and logarithm functions | ||
| exponential | 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 | |
