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.
  
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== Notation ==
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The derivative of f(x) can be expressed in several ways including:
  
== How to Find the Derivative ==
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* <math>\frac{d}{dx}</math>
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* <math>f'(x)</math>
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* <math>f'</math>
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== Finding the Derivative ==
  
 
For any constant, the derivative is 0.
 
For any constant, the derivative is 0.
  
For any monomial <math>nx</math>, the derivitave is n.
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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.
 
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>
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The [[chain rule]] states that the derivative of any <math>ax^n</math> is <math>anx^{n-1}</math>
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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>
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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.

Revision as of 22:04, 9 September 2006

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.

Notation

The derivative of f(x) can be expressed in several ways including:

  • $\frac{d}{dx}$
  • $f'(x)$
  • $f'$

Finding the Derivative

For any constant, the derivative is 0.

For any monomial $nx$, the derivative is n.

Note that when we take the derivative of any polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, 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 $ax^n$ is $anx^{n-1}$

To find the derivative of $f(x) \cdot g(x)$ we cannot do what we did with addition. We must instead use the product rule: $(f(x) \cdot g(x))' = f'g + g'f$

The quotient rule states that $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$

The following pages provide additional information on derivatives.

See also

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