# How do you differentiate f(x) = (5x)/(3x^2-x+1) using the quotient rule?

Jul 22, 2017

$f ' \left(x\right) = \frac{5 - 15 {x}^{2}}{3 {x}^{2} - x + 1} ^ 2$

#### Explanation:

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

$f ' \left(x\right) = \frac{h \left(x\right) g ' \left(x\right) - g \left(x\right) h ' \left(x\right)}{h \left(x\right)} ^ 2 \leftarrow \text{ quotient rule}$

$g \left(x\right) = 5 x \Rightarrow g ' \left(x\right) = 5$

$h \left(x\right) = 3 {x}^{2} - x + 1 \Rightarrow h ' \left(x\right) = 6 x - 1$

$\Rightarrow f ' \left(x\right) = \frac{\left(3 {x}^{2} - x + 1\right) .5 - 5 x \left(6 x - 1\right)}{3 {x}^{2} - x + 1} ^ 2$

$\textcolor{w h i t e}{\Rightarrow f ' \left(x\right)} = \frac{15 {x}^{2} - 5 x + 5 - 30 {x}^{2} + 5 x}{3 {x}^{2} - x + 1} ^ 2$

$\textcolor{w h i t e}{\Rightarrow f ' \left(x\right)} = \frac{5 - 15 {x}^{2}}{3 {x}^{2} - x + 1} ^ 2$