Derivative of the Natural Log Function



In this lesson, we will learn how to find the derivative of the natural log function (ln).

Related Topics: More Calculus Lessons

Natural Log (Ln)

The Natural Log is the logarithm to the base e. where e is an irrational constant approximately equal to 2.718281828. The natural logarithm is usually written ln(x) or loge(x).

The natural log is the inverse function of the exponential function. They are related by the following identities:

eln(x) = x

ln(ex) = x

You may want to look at the video below for a review of the exponential and natural log functions.

Derivative of ln(x)

Using the Chain Rule, we get



Example:

Differentiate y = ln(x2 +1)

Solution:

Using the Chain Rule, we get

 

Example:

Differentiate

Solution:

Videos

Derivatives of Logarithmic Functions
The derivative of the natural logarithmic function (ln[x]) is simply 1 divided by x. This derivative can be found using both the definition of the derivative and a calculator. Derivatives of logarithmic functions are simpler than they would seem to be, even though the functions themselves come from an important limit in Calculus.

This video gives the formulas for finding derivatives of logarithmic functions and use them to find derivatives.



Derivatives of Logarithmic Functions - More Examples





Derivatives of the Natural Log Function (Basic)
This video provides two basic examples of differentiating the natural logarithmic function.



Derivatives of the Natural Log Function with the Chain Rule
This video provides an example of differentiating the natural logarithmic function using the chain rule.







Derivatives of Natural Logarithms
How to differentiate the function y = ln(x), and some examples.



The Derivative of the Natural Log Function
We give two justifications for the formula for the derivative of the natural log function. If you want to see where this formula comes from, this is the video to watch.







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