Now that we have a little practice (more will be needed of course) in breaking up composite functions, we are ready to state the Chain Rule. Theorem If g is differentiable at x and f is differentiable at g(x), then the ...
In practice we often are interested in calculating the derivatives when the variable x is replaced by a function u(x). This requires the use of the chain rule. For example, d dx (sin −1 u) = 1 √ 1 −u 2 du dx = du dx √ 1 −u 2 , |u| < 1 The other functions ar...