Difference between revisions of "Derivative/Definition"
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<center>[[Image:derivative2.PNG]]</center> | <center>[[Image:derivative2.PNG]]</center> | ||
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<center><math> f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}h. </math></center> | <center><math> f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}h. </math></center> | ||
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+ | If this limit exists, it is the derivative of <math>f</math> at <math>x</math>. If it does not exist, we say that <math>f</math> is not differentiable at <math>x</math>. | ||
== See also == | == See also == | ||
* [[Calculus]] | * [[Calculus]] | ||
* [[Derivative]] | * [[Derivative]] |
Revision as of 11:47, 7 September 2006
The derivative of a function is defined as the instantaneous rate of change of the function at a certain point. For a line, this is just the slope. For more complex curves, we can find the rate of change between two points on the curve easily since we can draw a line through them.
In the image above, the rate of change between the two points is the slope of the line that goes through them: .
We can move the second point closer to the first one to find a more accurate value of the derivative. Thus, taking the limit as goes to 0 will give us the derivative of the function at
:
![$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}h.$](http://latex.artofproblemsolving.com/8/6/b/86bd376049080d6bd506d0e5955b97b874cfc33f.png)
If this limit exists, it is the derivative of at
. If it does not exist, we say that
is not differentiable at
.