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In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.
Suppose is a function of which is a function of (it is assumed that is differentiable at and, and is differentiable at. To prove the chain rule we use the definition of the derivative. We now multiply by and perform some algebraic manipulation. Note that as approaches 0, also approaches 0.