Derivative of the Natural Log Function

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 log_e(x).

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

e^ln(x) = x

ln(e^x) = 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.

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