# What is the quotient rule

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

The quotient rule is a formula for the derivative of the quotient of two functions, for which derivatives exist.

Equations

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

Then,

$$f'(x) = \frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}$$

here, $h(x) \: \neq \: 0$

Extended explanation

Even though a quotient can always be differentiated using the product and chain rules, it is easier and more efficient to remember and use the quotient rule.

Proof of the quotient rule:
$$f(x) = \frac{g(x)}{h(x)} = g(x)[h(x)]^{-1}$$
Using the product and chain rules:
$$f'(x) = g'(x)\:[h(x)]^{-1} - \: [h(x)]^{-2} \: h'(x)\:g(x)$$
and, putting this over a common denominator:
$$f'(x) = \frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}$$

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