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Chain Rule for Differentiation

For two differentiable functions

and

, the derivative of the composite function

is given by:

or, equivalently:

If the function

is differentiable:

If , where and are constants, then:
If , where is an integer, then:

Derivative of the Logarithmic Function

If , then:
If , then:

For a differentiable function

, where

, if

, then:

When finding the derivative of a function

, if the function has variables in both the base and the exponent, or is a complicated fraction, you can take the natural logarithm of both sides and differentiate with respect to

to find the derivative.

Derivative of the Function

is a real number and

, then: