How Do You Find A Derivative At A Point From the Definition?

The most difficult part of finding a derivative is evaluating the limit involved in the definition of the derivative at a point. Often there is some algebra and simplifying involved as the example below demonstrates.

Problem Suppose the function g(x) is given by

Use the definition of the derivative at a point to compute g´(3).

Solution The definition of the derivative of g(x) at x = 3 is

The function value g(3) is calculated to be

The function value g(3 + h) must be calculated carefully.

Form the difference quotient and simplify:

The derivative is completed by taking the limit as h approaches zero,

The derivative of g(x) evaluated at x = 3, g´(3), is 11.