| Line 18: | Line 18: | ||
| Please continue | | Please continue | ||
| write a rule here | | write a rule here | ||
| + | |- | ||
| + | ! 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 | ||
| + | |- | ||
| + | | first order | ||
| + | | <math>\frac{d}{dx}\left( u v \right)= u \frac{dv }{dx} + v \frac{du }{dx} </math> | ||
| + | |- | ||
| + | | second order | ||
| + | | <math>\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} </math> | ||
| + | |- | ||
| + | | 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> | ||
| + | |- | ||
| + | | n-th order | ||
| + | | <math>\frac{d^n}{dx^n}\left( u v \right)= u \frac{d^n v }{dx^n} + </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 | ||
|- | |- | ||
| − | | | + | | hyperbolic sine |
| − | | < | + | | <math>\text{sh } u</math> |
| − | | <math> | + | | <math>\text{ch } u \frac{du}{dx}</math> |
|- | |- | ||
| | | | ||
Revision as of 08:22, 26 October 2010
Table of Derivatives
| Laplace Transform Pairs and Properties | |
|---|---|
| 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} $ |
| n-th order | $ \frac{d^n}{dx^n}\left( u v \right)= u \frac{d^n v }{dx^n} + $ |
| Derivatives of trigonometric functions | ||
|---|---|---|
| sine | 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 | |
