Product Rule - Proof of The Product Rule

Proof of The Product Rule

A rigorous proof of the product rule can be given using the properties of limits and the definition of the derivative as a limit of Newton's difference quotient.

If

and ƒ and g are each differentiable at the fixed number x, then

Now the difference

is the area of the big rectangle minus the area of the small rectangle in the illustration.

The region between the smaller and larger rectangle can be split into two rectangles, the sum of whose areas is

Therefore the expression in (1) is equal to

Assuming that all limits used exist, (4) is equal to

\left(\lim_{w\to x}f(x)\right) \left(\lim_{w\to x} {g(w) - g(x) \over w - x}\right) + \left(\lim_{w\to x} g(w)\right) \left(\lim_{w\to x} {f(w) - f(x) \over w - x} \right). \qquad\qquad(5)

Now

This holds because f(x) remains constant as w → x.

This holds because differentiable functions are continuous (g is assumed differentiable in the statement of the product rule).

Also:

and

because f and g are differentiable at x;

We conclude that the expression in (5) is equal to

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