# How do you use the quotient rule to differentiate y= (x^2+4x)/(x+2)^2?

Jun 3, 2016

$\frac{8}{x + 2} ^ 3$

#### Explanation:

differentiate using the $\textcolor{b l u e}{\text{quotient rule}}$

$f \left(x\right) = \frac{g \left(x\right)}{h \left(x\right)} \text{ then } f ' \left(x\right) = \frac{h \left(x\right) . g ' \left(x\right) - g \left(x\right) h ' \left(x\right)}{h \left(x\right)} ^ 2$
$\text{---------------------------------------------------------------------}$
$g \left(x\right) = {x}^{2} + 4 x \Rightarrow g ' \left(x\right) = 2 x + 4$

$h \left(x\right) = {\left(x + 2\right)}^{2} \Rightarrow h ' \left(x\right) = 2 \left(x + 2\right)$
$\text{-----------------------------------------------------------------------------}$
Substitute these values into f'(x)

f'(x)=((x+2)^2(2x+4)-(x^2+4x)2(x+2))/((x+2)^4

$= \frac{\left(x + 2\right) \left[\left(x + 2\right) \left(2 x + 4\right) - 2 \left({x}^{2} + 4 x\right)\right]}{x + 2} ^ 4$

=(cancel((x+2))[2x^2+4x+4x+8-2x^2-8x])/(cancel((x+2))(x+2)^3

$= \frac{8}{x + 2} ^ 3$