# Quotient rule review

Review your knowledge of the Quotient rule for derivatives, and use it to solve problems.

## What is the Quotient rule?

The Quotient rule tells us how to differentiate expressions that are the quotient of two other, more basic, expressions:
$\dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right]=\dfrac{\dfrac{d}{dx}[f(x)]\cdot g(x)-f(x)\cdot\dfrac{d}{dx}[g(x)]}{[g(x)]^2}$
Basically, you take the derivative of $f$ multiplied by $g$, subtract $f$ multiplied by the derivative of $g$, and divide all that by $[g(x)]^2$.

## What problems can I solve with the Quotient rule?

### Example 1

Consider the following differentiation of $\dfrac{\sin(x)}{x^2}$:
\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(\dfrac{\sin(x)}{x^2}\right) \\\\ &=\dfrac{\dfrac{d}{dx}(\sin(x))x^2-\sin(x)\dfrac{d}{dx}(x^2)}{(x^2)^2}&&\gray{\text{Quotient rule}} \\\\ &=\dfrac{\cos(x)\cdot x^2-\sin(x)\cdot 2x}{(x^2)^2}&&\gray{\text{Differentiate }\sin(x)\text{ and }x^2} \\\\ &=\dfrac{x\left(x\cos(x)-2\sin(x)\right)}{x^4}&&\gray{\text{Simplify}} \\\\ &=\dfrac{x\cos(x)-2\sin(x)}{x^3}&&\gray{\text{Cancel common factors}} \end{aligned}

Problem 1
$f(x)=\dfrac{x^2}{e^x}$
$f'(x)=$

Want to try more problems like this? Check out this exercise.

### Example 2

Suppose we are given this table of values:
$x$$f(x)$$g(x)$$f'(x)$$g'(x)$
$4$$-4$$-2$$0$$8$
$H(x)$ is defined as $\dfrac{f(x)}{g(x)}$, and we are asked to find $H'(4)$.
The Quotient rule tells us that $H'(x)$ is $\dfrac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}$. This means $H'(4)$ is $\dfrac{f'(4)g(4)-f(4)g'(4)}{[g(4)]^2}$. Now let's plug the values from the table in the expression:
\begin{aligned} H'(4)&=\dfrac{f'(4)g(4)-f(4)g'(4)}{[g(4)]^2} \\\\ &=\dfrac{(0)(-2)-(-4)(8)}{(-2)^2} \\\\ &=\dfrac{32}{4} \\\\ &=8 \end{aligned}

$x$$g(x)$$h(x)$$g'(x)$$h'(x)$
$-2$$4$$1$$-1$$2$
$F(x)=\dfrac{g(x)}{h(x)}$
$F'(-2)=$