Difference between revisions of "Derivative"

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The '''derivative''' of a [[function]] is defined as the instantaneous [[rate]] of change of the function with respect to one of the [[variable]]s.  Note that not every function has a derivative.
 
The '''derivative''' of a [[function]] is defined as the instantaneous [[rate]] of change of the function with respect to one of the [[variable]]s.  Note that not every function has a derivative.
 
== Notation ==
 
The derivative of f(x) can be expressed in several ways including:
 
 
* <math>\frac{d}{dx}</math>
 
* <math>f'(x)</math>
 
* <math>f'</math>
 
 
== Finding the Derivative ==
 
 
For any constant, the derivative is 0.
 
 
For any monomial <math>nx</math>, the derivative is n.
 
 
Note that when we take the derivative of any polynomial <math>a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0</math>, we can take the derivative of each addend and then add these to find the derivative of the polynomial.
 
 
The [[chain rule]] states that the derivative of any <math>ax^n</math> is <math>anx^{n-1}</math>
 
 
To find the derivative of <math>f(x) \cdot g(x)</math> we cannot do what we did with addition.  We must instead use the [[product rule]]: <math>(f(x) \cdot g(x))' = f'g + g'f</math>
 
 
The [[quotient rule]] states that <math>(\frac{f}{g})' = \frac{f'g - fg'}{g^2}</math>
 
  
 
The following pages provide additional information on derivatives.
 
The following pages provide additional information on derivatives.
* [[Derivative/Notations | Notations]]
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* [[Derivative/Definition | Formal definition]] of the derivative and examples of applications
* [[Derivative/Formal definition | Formal definition]] of the derivative
 
 
* [[Derivative/Formulas | Formulas]]
 
* [[Derivative/Formulas | Formulas]]
  
 
== See also ==
 
== See also ==
 
* [[Calculus]]
 
* [[Calculus]]
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* [[Differentiation Rules]]
 
* [[Integral]]
 
* [[Integral]]
 
* [[Chain Rule]]
 
* [[Chain Rule]]
  
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[[Category:Calculus]]

Latest revision as of 19:26, 3 March 2010

The derivative of a function is defined as the instantaneous rate of change of the function with respect to one of the variables. Note that not every function has a derivative.

The following pages provide additional information on derivatives.

See also

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