Difference between revisions of "Derivative"

(added displaystyle tags)
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* <math>\frac{d}{dx}</math>
 
* <math>\frac{d}{dx}</math>
* <math>f'(x)</math>
+
* <math>\displaystyle f'(x)</math>
* <math>f'</math>
+
* <math>\displaystyle f'</math>
  
 
== Finding the Derivative ==
 
== Finding the Derivative ==
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For any monomial <math>nx</math>, the derivative is n.
 
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>\displaystyle 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>
+
The [[chain rule]] states that the derivative of any <math>\displaystyle ax^n</math> is <math>\displaystyle 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>
+
To find the derivative of <math>\displaystyle f(x) \cdot g(x)</math> we cannot do what we did with addition.  We must instead use the [[product rule]]: <math>\displaystyle (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 [[quotient rule]] states that <math>\displaystyle (\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 22:05, 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:

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

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 $\displaystyle a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, 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 $\displaystyle ax^n$ is $\displaystyle anx^{n-1}$

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

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

The following pages provide additional information on derivatives.

See also

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