# Use the quotient rule to find the derivative?

## $f \left(x\right) = \frac{\left(4 x - 1\right) \left(3 {x}^{2} + 1\right)}{5 x + 2}$

Mar 8, 2018

$\frac{d}{\mathrm{dx}} \left(g \frac{x}{h \left(x\right)}\right) = \frac{g ' \left(x\right) \cdot h \left(x\right) - h ' \left(x\right) \cdot g \left(x\right)}{{h}^{2} \left(x\right)}$

#### Explanation:

Using the quotient rule,
we get $g \left(x\right) = \left(4 x - 1\right) \left(3 {x}^{2} + 1\right)$ and $h \left(x\right) = 5 x + 2$

$g ' \left(x\right) = 4 \left(3 {x}^{2} + 1\right) + 6 x \left(4 x - 1\right)$
$h ' \left(x\right) = 5$

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{\left(4 \left(3 {x}^{2} + 1\right) + 6 x \left(4 x - 1\right)\right) \left(5 x + 2\right) - 5 \left(4 x - 1\right) \left(3 {x}^{2} + 1\right)}{5 x + 2} ^ 2$

$\implies \frac{120 {x}^{3} + 57 {x}^{2} - 12 x + 13}{5 x + 2} ^ 2$