What is the product rule

1. Jul 23, 2014

Greg Bernhardt

Definition/Summary

The product rule is a method for finding the derivative of a product of functions.

Equations

$$(fg)'\ =\ f'g\ +\ fg'$$

$$(fgh)'\ =\ f'gh\ +\ fg'h\ +\ fgh'$$

Extended explanation

If a function F is the product of two other functions f and g (i.e. F(x) = f(x)g(x)), then the product rules states that:

$$\frac{d}{dx}F \ = \ \frac{df}{dx}g \ + \ f\frac{dg}{dx}$$

Proof:

$$\frac{F(x + h) - F(x)}{h} \ = \ \frac{f(x + h)g(x + h) - f(x)g(x)}{h}$$

$$= \ \frac{f(x + h)g(x + h) \ - \ f(x)g(x + h) \ + \ f(x)g(x + h) \ - \ f(x)g(x)}{h}$$

$$= \ \frac{f(x + h) - f(x)}{h}g(x + h) \ + \ f(x)\frac{g(x+h) - g(x)}{h}$$

Now take the limit as h approaches zero.

$$\frac{d}{dx}F \ = \ \lim_{h \to 0}\frac{F(x + h) - F(x)}{h} \ = \ \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}g(x + h) \ + \ \lim_{h \to 0}f(x)\frac{g(x+h) - g(x)}{h} \ = \ \frac{df}{dx}g \ + \ f\frac{dg}{dx}$$

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