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.

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.