# Quotient rule

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let $f(x)=g(x)/h(x),$ where both $g$ and $h$ are differentiable and $h(x)\neq 0.$ The quotient rule states that the derivative of $f(x)$ is

$f'(x)={\frac {[h(x)]^}}.$ ## Examples

1. A basic example:
${\begin{\frac }{\frac }}}&={\frac {\left({\frac }e^\right)(x^)-(e^)\left({\frac }x^\right)}{(x^)^}}\\&={\frac {(e^)(x^)-(e^)(2x)}}}\\&={\frac (x-2)}}}.\end}$ 2. The quotient rule can be used to find the derivative of $f(x)=\tan x={\tfrac {\sin x}{\cos x}}$ as follows.
${\begin{\frac }\tan x&={\frac }{\frac {\sin x}{\cos x}}\\&={\frac {\left({\frac }\sin x\right)(\cos x)-(\sin x)\left({\frac }\cos x\right)}{\cos ^x}}\\&={\frac {\cos ^x+\sin ^x}{\cos ^x}}\\&={\frac {\cos ^x}}=\sec ^x.\end}$ ## Proofs

### Proof from derivative definition and limit properties

Let $f(x)=g(x)/h(x).$ Applying the definition of the derivative and properties of limits gives the following proof.

${\beginf'(x)&=\lim _{\frac }\\&=\lim _{\frac {{\frac }-{\frac }}}\\&=\lim _{\frac }\\&=\lim _{\frac }\cdot \lim _{\frac }\\&=\left(\lim _{\frac }\right)\cdot {\frac }}\\&=\left(\lim _{\frac }-\lim _{\frac }\right)\cdot {\frac }}\\&=\left(h(x)\lim _{\frac }-g(x)\lim _{\frac }\right)\cdot {\frac }}\\&={\frac }}.\end}$ ### Proof using implicit differentiation

Let $f(x)={\frac },$ so $g(x)=f(x)h(x).$ The product rule then gives $g'(x)=f'(x)h(x)+f(x)h'(x).$ Solving for $f'(x)$ and substituting back for $f(x)$ gives:

${\beginf'(x)&={\frac }\\&={\frac }\cdot h'(x)}}\\&={\frac }}.\end}$ ### Proof using the chain rule

Let $f(x)={\frac }=g(x)h(x)^{-1}.$ Then the product rule gives

$f'(x)=g'(x)h(x)^{-1}+g(x)\cdot {\frac }(h(x)^{-1}).$ To evaluate the derivative in the second term, apply the power rule along with the chain rule:

$f'(x)=g'(x)h(x)^{-1}+g(x)\cdot (-1)h(x)^{-2}h'(x).$ Finally, rewrite as fractions and combine terms to get

${\beginf'(x)&={\frac }-{\frac }}\\&={\frac }}.\end}$ ## Higher order formulas

Implicit differentiation can be used to compute the nth derivative of a quotient (partially in terms of its first n − 1 derivatives). For example, differentiating $fh=g$ twice (resulting in $f''h+2f'h'+fh''=g''$ ) and then solving for $f''$ yields

$f''=\left({\frac }\right)''={\frac }.$ 