**Limit of the difference quotient**

In 'The rate of change of a function' section, it was found that the average rate of change between *[a, b]* is

This allows us to find the instantaneous rate of change at point *a* when we take number *b* extremely close to that of number *a*. That is, we get the limit value when *b* approaches the number *a.*

**Example 1**

Find the instantaneous rate of change of the function* f * when *x = 4*

Select point *B* from the function graph, which is after point *A*. Denote this point with the letter* x*. The *y*-coordinate of the point is then *f(x)*.

Bring the point as close as possible to point *A*

Calculate a few values of the function with different values of *x* near value *4*.

As* x* approaches *4*, the rate of change values would appear to be approaching *6*. We find the limit for rate of change when *x* approaches *4*.

Then we get the slope of the tangent in *4*

The instantaneous rate of change when *x = 4 *is *6*. This is the derivative of the function when* x = 4.*

The derivative of the function *f* in *a*

If a limit value exists, the function has a derivative in *a*. This is called the difference quotient limit value.

**Example 2**

Find the derivative of the function *g* when *x = 2.*

Solution

Find the difference quotient limit value in *2*

The derivative of the function *g* at *2 *is *-1.*