# Derivatives Of Logarithmic Functions

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Brief Review of Exponential Functions and Logarithmic Functions

An exponential function is a function of the form:

where is a non-zero number and is a variable.

The logarithmic function is the inverse function of the exponential function. The logarithmic function is of the form:

where the base is a non-zero number and is a variable.

If we have the exponential function of with Euler’s constant of then the corresponding inverse would be .

Derivatives of Logarithmic Functions – Formulas

Given that is of the form then the derivative is:

A more simpler case is when we have . The derivative would be:

The most simplest and most common case is taking the derivative of .

(Note that ln(e) = 1.)

Examples

Here are some examples of differentiating logarithmic functions.

Example One:

Given that then the derivative is:

Example Two:

If we have then the derivative is:

Example Three:

The derivative of the function is:

Example Four:

This last example is more involved. Suppose that the function is . At first, you may be terrified. However, if you look at this more closely this is a case of using chain rule. You would need to differentiate multiple times. The derivative of is as follows.

When dealing with chain rule cases, go from the outside to the inside. From the first line to the second line the derivative of is . Then we take the derivative of the inside which is .

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