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.