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3.20 Chain rule

The chain rule is used to differentiate composite functions.

Chain Rule

Let ff and gg be differentiable functions and let s(x)=g(f(x))s(x)=g(f(x)). Then

s(x)=g(f(x))f(x).s^{\prime}(x)=g^{\prime}(f(x))f^{\prime}(x).

To differentiate a composition of functions, differentiate all of the functions in the composition one at a time, starting from the outside and multiplying by each new derivative thus formed. Work out the derivative at the same point that the function was originally worked out.