# How do you differentiate f(x)= (4 x^2 + 5x -8 )/ (x- 1 ) using the quotient rule?

Jan 13, 2017

$f ' \left(x\right) = \frac{4 {x}^{2} - 8 x + 3}{x - 1} ^ 2$

#### Explanation:

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

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

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

$\text{here } g \left(x\right) = 4 {x}^{2} + 5 x - 8 \Rightarrow g ' \left(x\right) = 8 x + 5$

$\text{and } h \left(x\right) = x - 1 \Rightarrow h ' \left(x\right) = 1$

$\Rightarrow f ' \left(x\right) = \frac{\left(x - 1\right) \left(8 x + 5\right) - \left(4 {x}^{2} + 5 x - 8\right) .1}{x - 1} ^ 2$

$= \frac{8 {x}^{2} - 3 x - 5 - 4 {x}^{2} - 5 x + 8}{x - 1} ^ 2$

$= \frac{4 {x}^{2} - 8 x + 3}{x - 1} ^ 2$