# AP Calculus BC : Instantaneous Rate of Change, Average Rate of Change, and Linear Approximation

## Example Questions

### Example Question #1 : Derivative At A Point

Calculate the derivative of  at the point .

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of .

Then, replace the value of x with the given point.

For example, if , then we are looking for the value of , or the derivative of  at .

Calculate

Derivative rules that will be needed here:

• Derivative of a constant is 0. For example,
• Taking a derivative on a term, or using the power rule, can be done by doing the following:

Then, plug in the value of x and evaluate

### Example Question #1 : Instantaneous Rate Of Change, Average Rate Of Change, And Linear Approximation

Evaluate the first derivative if

and .

Explanation:

First we must find the first derivative of the function.

Because the derivative of the exponential function is the exponential function itelf, or

and taking the derivative is a linear operation,

we have that

Now setting

Thus

### Example Question #1 : Instantaneous Rate Of Change, Average Rate Of Change, And Linear Approximation

Find the rate of change of f(x) when x=3.

Explanation:

Find the rate of change of f(x) when x=3.

To find a rate of change, we need to find the derivative.

First, recall the following rules:

We can apply these two derivative rules to our function to get  our first derivative. Then we need to plug in 3 for x and solve.