# 7.2: The Logarithmic Derivative

*This activity is intended to supplement Calculus, Chapter 6, Lesson 3.*

## Problem 1 – The Derivative of

If is a point on and is the inverse of , then is a point on . We know that and , so is a point on and is a point on . We could do this for several points and keep getting the same inverse results.

Thus, if , then will be equivalent to because they are inverses of one another. Now we can take the implicit derivative with respect to of .

Use the **limit** command to test this formula. Be careful with your parentheses.

- Find .
- Do the same with .
- What is

Use the **derivative** command to find the derivative of the logarithmic function .

## Problem 2 – The Derivative of

What happens if our logarithm has a base other than ? We need to know how to take the derivative of the function .

First we want to compare and .

To enter , use the alpha keys to spell out **log**.

Within the parentheses, enter the expression, then the base.

- Graph both functions ( and ) on the same set of axes. Sketch your graph to the right. What do you notice?

- Do the same steps with and . What do you notice?

- Simplify the following ratios.

Sometimes the ratio is written as . We can rewrite this ratio as and call it an identity.

- Graph the following functions on the same set of axes: . What was the result?

What happens when we take the derivative of . Use the **derivative** command to find the derivatives of the functions below.

- Do you notice a pattern?

What does equal? If we use the formula from earlier in this class, we get .

Therefore, the general result is .

## Problem 3 – Derivative of Exponential and Logarithmic Functions Using the Chain Rule

Now we want to take the derivative of more complicated functions:

Recall: where depends on .

- Suppose that , where depends on . Using the chain rule, take the derivative of this function.

Find the derivative of the following functions with the chain rule.

Identify and for each function before you find the derivative.