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Some of the content of this guide was modeled after a guide originally created by Openstax and has been adapted for the GPRC Learning Commons in September 2021. The graphs are generated using Desmos. This work is licensed under a Creative Commons BY 4.0 International License.

Definition of a Derivative

A derivative is used to find the slope of a curve at any point along that curve.  The function f'(a) is read as "f prime of a" and can be found using either definition of a limit, either as x approaches a or as h approaches 0.


Below are two examples showing one function being solved with both definitions.  The key to solving these functions is to isolate a factor of the denominator in the numerator and cancel those factors out so the function is no longer dividing by zero.  For x approaching a, isolate the term (x-a) in the numerator; or for h approaching 0, isolate the term h in the numerator.

Example 1:


Example 2:

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