Differentiation rules > Chain rule
123456Chain rule

Theory

A composed function is a function consisting of two or more functions that are connected in series.

The derivative of a composed function can be found with

Differentiation rule 4 (chain rule):
If S ( x ) = f ( g ( x ) ) then S ' ( x ) = f ' ( g ( x ) ) g ' ( x ) .

You can use this rule to prove the power rule for root functions:

If f ( x ) = x n = x 1 n then f ' ( x ) = 1 n x 1 n 1 for positive integer n and x 0 .

In general:

Differentiation rule 5 (general power rule):
If f ( x ) = x r then f ' ( x ) = r x r 1 for any real value of r .

To be able to prove the differentiation rule, you first need to learn how to differentiate exponential functions and logaritmic functions. That will come later.

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