Proof of the Chain Rule

Techniques of Differentiation Topic Summary
Review Exercises on Techniques Index of Calculus Proofs Student Home Everything for Calculus Epañol 
The Chain Rule
If the function $f$ has derivative $f'$ and the function $u$ has derivative $\frac{du}{dx}$, then the composite function $f(u)$ is differentiable, and

Proof From the definition of the derivative,
Putting (III) and (V) together now gives
which is the chain rule.