Quotient Rule (Derivative) | Math Rules

Q: What is an example of the Quotient Rule (Derivative)?

To find the derivative of $f(x) = \frac{\sin x}{x}$, let $u(x) = \sin x$ and $v(x) = x$. Then $u'(x) = \cos x$ and $v'(x) = 1$. Applying the rule: $f'(x) = \frac{(\cos x)(x) - (\sin x)(1)}{x^2} = \frac{x \cos x - \sin x}{x^2}$.

The rule

Description

The Quotient Rule is a fundamental formula in differential calculus used to find the derivative of a function that is expressed as the ratio of two differentiable functions. It states that if a function $f (x)$ is given by $f (x) = \frac{u ( x )}{v ( x )}$ , where $u (x)$ and $v (x)$ are differentiable functions and $v (x) \neq = 0$ , then its derivative $f^{'} (x)$ is given by the formula $f^{'} (x) = \frac{u ^{'} ( x ) v ( x ) - u ( x ) v ^{'} ( x )}{( v ( x ) ) ^{2}}$ .

See it in action

Examples

1

To find the derivative of $f (x) = \frac{s i n x}{x}$ , let $u (x) = sin x$ and $v (x) = x$ . Then $u^{'} (x) = cos x$ and $v^{'} (x) = 1$ . Applying the rule: $f^{'} (x) = \frac{( c o s x ) ( x ) - ( s i n x ) ( 1 )}{x ^{2}} = \frac{x c o s x - s i n x}{x ^{2}}$ .

2

To find the derivative of $g (x) = \frac{e ^{x}}{x ^{2}}$ , let $u (x) = e^{x}$ and $v (x) = x^{2}$ . Then $u^{'} (x) = e^{x}$ and $v^{'} (x) = 2 x$ . Applying the rule: $g^{'} (x) = \frac{( e ^{x} ) ( x ^{2} ) - ( e ^{x} ) ( 2 x )}{( x ^{2} ) ^{2}} = \frac{x ^{2} e ^{x} - 2 x e ^{x}}{x ^{4}} = \frac{x e ^{x} ( x - 2 )}{x ^{4}} = \frac{e ^{x} ( x - 2 )}{x ^{3}}$ .

Good to know

Key Facts

The Quotient Rule is essential for differentiating rational functions and other functions expressed as quotients.
It can be derived from the Product Rule and the Chain Rule by writing $\frac{u}{v}$ as $u \cdot v^{- 1}$ .

Common questions

Frequently Asked Questions

The Quotient Rule is a fundamental formula in differential calculus used to find the derivative of a function that is expressed as the ratio of two differentiable functions. It states that if a function $f (x)$ is given by $f (x) = \frac{u ( x )}{v ( x )}$ , where $u (x)$ and $v (x)$ are differentiable functions and $v (x) \neq = 0$ , then its derivative $f^{'} (x)$ is given by the formula $f^{'} (x) = \frac{u ^{'} ( x ) v ( x ) - u ( x ) v ^{'} ( x )}{( v ( x ) ) ^{2}}$ .

To find the derivative of $f (x) = \frac{s i n x}{x}$ , let $u (x) = sin x$ and $v (x) = x$ . Then $u^{'} (x) = cos x$ and $v^{'} (x) = 1$ . Applying the rule: $f^{'} (x) = \frac{( c o s x ) ( x ) - ( s i n x ) ( 1 )}{x ^{2}} = \frac{x c o s x - s i n x}{x ^{2}}$ .

The Quotient Rule is essential for differentiating rational functions and other functions expressed as quotients. It can be derived from the Product Rule and the Chain Rule by writing $\frac{u}{v}$ as $u \cdot v^{- 1}$ .