# How do you differentiate f(x)= ( 13x^2+ 3x ) / ( 4e^x + 2 )  using the quotient rule?

##### 1 Answer
Nov 24, 2016

$f ' \left(x\right) = \frac{{e}^{x} \left(92 x - 52 {x}^{2} + 12\right) + 52 x + 6}{4 {e}^{x} + 2} ^ 2$

#### Explanation:

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

$\text{If" f(x)=(g(x))/(h(x))" then}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{f ' \left(x\right) = \frac{h \left(x\right) g ' \left(x\right) - g \left(x\right) h ' \left(x\right)}{h \left(x\right)} ^ 2}} \textcolor{w h i t e}{\frac{2}{2}} |}}$

Here $g \left(x\right) = 13 {x}^{2} + 3 x \Rightarrow g ' \left(x\right) = 26 x + 3$

and $h \left(x\right) = 4 {e}^{x} + 2 \Rightarrow h ' \left(x\right) = 4 {e}^{x}$

$\Rightarrow f ' \left(x\right) = \frac{\left(4 {e}^{x} + 2\right) \left(26 x + 3\right) - \left(13 {x}^{2} + 3 x\right) .4 {e}^{x}}{4 {e}^{x} + 2} ^ 2$

$= \frac{104 x {e}^{x} + 12 {e}^{x} + 52 x + 6 - 52 {x}^{2} {e}^{x} - 12 x {e}^{x}}{4 {e}^{x} + 2} ^ 2$

$= \frac{92 x {e}^{x} - 52 {x}^{2} {e}^{x} + 12 {e}^{x} + 52 x + 6}{4 {e}^{x} + 2} ^ 2$

$= \frac{{e}^{x} \left(92 x - 52 {x}^{2} + 12\right) + 52 x + 6}{4 {e}^{x} + 2} ^ 2$