Product Rule in differentiation

Product Rule

If f(x) and g(x) are differentiable, then

Proof

The proof of the product rule using the first principle seems easier by studying the following diagram:

Since the area of the biggest rectangle subtract the area of pink rectangle is equal to the area of blue rectangle and the area of the purple rectangle, we have:

f (x +Dx) g (x +Dx) – f(x) g(x) = f (x +Dx)[g (x +Dx) - g(x)] + [f (x +Dx) - f(x)] g(x)

Divide both sides by Dx and take the limit Dx ® 0, we have:

Ha, the product rule is proved: