(17 intermediate revisions by 6 users not shown)
Line 1: Line 1:
 +
[[Category:Formulas]]
 +
 +
keywords:quotient rule, chain rule, Leibniz rule
 +
 +
<center><font size= 4>
 +
'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
 +
</font size>
 +
 +
'''Derivatives'''
 +
 +
click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
 +
 +
</center>
 +
 +
----
 +
 +
 
{|
 
{|
−
|-
 
−
! style="background-color: rgb(228, 188, 126); 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" | Table of Derivatives
 
 
|-
 
|-
 
! 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
Line 14: Line 29:
 
| <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>
 
|-
 
|-
−
| Please continue
+
| Quotient rule
−
| write a rule here
+
|<math> \frac{d}{dx} ( \frac{u}{v} ) = \frac{v ( \frac{du}{dx} ) - u ( \frac{dv}{dx} )}{v^2}</math>
 +
|-
 +
| 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>
 +
|-
 +
|
 +
|<math> \frac{du}{dx} = \frac{1}{\frac{dx}{du}}</math>
 +
|-
 +
|
 +
|<math> \frac{dy}{dx} = \frac{dy}{du}/\frac{dx}{du}</math>
 
|-
 
|-
 
! 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
Line 26: Line 53:
 
|-
 
|-
 
| 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> [[User:Mboutinformulas|credit]]
+
| <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  
 
| n-th order  
Line 54: Line 81:
 
|-
 
|-
 
| <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>
−
|- 
 
−
| add function here
 
−
|  derivative 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="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
 
|-
 
|-
−
| exponential
+
| <math> \frac{d}{dx} \log_a u = \frac{log_a e}{u} \frac{du}{dx} \qquad a \neq 0,1</math>
−
| <span class="texhtml">''e''<sup>''u''</sup></span>
+
−
| <math>e^u \frac{du}{dx}</math>
+
 
|-
 
|-
−
|
+
| <math> \frac{d}{dx} \ln u = \frac{d}{dx} log_e u = \frac{1}{u} \frac{du}{dx}</math>
−
| add function here
+
|-
−
| derivative here
+
| <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>
 +
|-
 +
| <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>
 
|-
 
|-
 
! 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> \frac{d}{dx} \sinh u = \cosh u \frac{du}{dx}</math>
−
| <math>\text{sh } u</math>
+
|-
−
| <math>\text{ch } u \frac{du}{dx}</math>
+
| <math> \frac{d}{dx} \cosh u = \sinh u \frac{du}{dx}</math>
 +
|-
 +
| <math> \frac{d}{dx} \tanh u = \frac{1}{\cosh^2 u} \frac{du}{dx}</math>
 +
|-
 +
| <math> \frac{d}{dx} \coth u = - \frac{1}{\sinh^2 u} \frac{du}{dx}</math>
 +
|-
 +
| <math> \frac{d}{dx} \frac{1}{\cosh u} = - \frac{\tanh u}{\cosh u} \frac{du}{dx} </math>
 +
|-
 +
| <math> \frac{d}{dx} \frac{1}{\sinh u} = - \frac{\coth u}{\sinh u} \frac{du}{dx} </math>
 +
|-
 +
| <math> \frac{d}{dx}\ \operatorname{arsinh}\ u = \frac{1}{\sqrt{u^2+1}} \frac{du}{dx}</math>
 +
|-
 +
| <math> \frac{d}{dx}\ \operatorname{arcosh}\ u = \frac{1}{\sqrt{u^2-1}} \frac{du}{dx}</math>
 +
|-
 +
| <math> \frac{d}{dx}\ \operatorname{artanh}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ -1 < u < 1 \ ) </math>
 +
|-
 +
| <math> \frac{d}{dx}\ \operatorname{arcoth}\ u = \frac{1}{1-u^2} \frac{du}{dx} \qquad ( \ u > 1 \ or \ u < -1 \ ) </math>
 
|-
 
|-
−
| 
 
−
| add function here
 
−
|  derivative here
 
 
|}
 
|}
  
  
 
[[Collective_Table_of_Formulas|Back to Collective Table of Formulas]]
 
[[Collective_Table_of_Formulas|Back to Collective Table of Formulas]]
−
[[Category:Formulas]]
 

Latest revision as of 17:40, 27 February 2015


keywords:quotient rule, chain rule, Leibniz rule

Collective Table of Formulas

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 \ ) $


Back to Collective Table of Formulas

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=Table_of_derivatives&oldid=69280"
Shortcuts
Help Main Wiki Page Random Page Special Pages Log in
Search

Alumni Liaison

has a message for current ECE438 students.

Sean Hu, ECE PhD 2009