Difference between revisions of "Derivative/Definition"
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| − | 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. | + | 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. |
<center>[[Image:derivative1.PNG]]</center> | <center>[[Image:derivative1.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> | ||
| − | 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>. | + | 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 08:36, 24 September 2007
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 
:
If this limit exists, it is the derivative of 
at . If it does not exist, we say that is not differentiable at .
