# How do you find the derivative of y = (2x^4 - 3x) / (4x - 1)?

May 31, 2016

Using the derivative rules we find that the answer is $\frac{24 {x}^{4} - 8 {x}^{3} + 3}{4 x - 1} ^ 2$

#### Explanation:

Derivative rules we need to use here are:
a. Power rule
b. Constant Rule
c. Sum and difference rule
d. Quotient rule

1. Label and derive the numerator and denominator
$f \left(x\right) = 2 {x}^{4} - 3 x$
$g \left(x\right) = 4 x - 1$

By applying the Power rule, constant rule, and sum and difference rules, we can derive both of these functions easily:
${f}^{'} \left(x\right) = 8 {x}^{3} - 3$
${g}^{'} \left(x\right) = 4$

at this point we will use the Quotient rule which is:
${\left[\frac{f \left(x\right)}{g \left(x\right)}\right]}^{'} = \frac{{f}^{'} \left(x\right) g \left(x\right) - f \left(x\right) {g}^{'} \left(x\right)}{g \left(x\right)} ^ 2$

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

From here you can simplify it to:
$\frac{24 {x}^{4} - 8 {x}^{3} + 3}{4 x - 1} ^ 2$

Thus the derivitive is the simplified answer.