# What is the derivative of x^3 * (2/3x^2 -1)^4?

Jan 13, 2017

$\frac{176}{81} {x}^{10} - \frac{32}{3} {x}^{8} + \frac{56}{3} {x}^{6} - \frac{40}{3} {x}^{4} + 3 {x}^{2}$

#### Explanation:

Expand the expression as

${x}^{3} \left[{\left(\frac{2}{3} {x}^{2}\right)}^{4} + 4 {\left(\frac{2}{3} {x}^{2}\right)}^{3} \left(- 1\right) + 6 {\left(\frac{2}{3} {x}^{2}\right)}^{2} {\left(- 1\right)}^{2} + 4 \left(\frac{2}{3} {x}^{2}\right) {\left(- 1\right)}^{3} + {\left(- 1\right)}^{4}\right]$

=$\frac{16}{81} {x}^{11} - \frac{32}{27} {x}^{9} + \frac{24}{9} {x}^{7} - \frac{8}{3} {x}^{5} + {x}^{3}$

Its first derivative would be

$\frac{176}{81} {x}^{10} - \frac{32}{3} {x}^{8} + \frac{56}{3} {x}^{6} - \frac{40}{3} {x}^{4} + 3 {x}^{2}$

Alternatively, product rule of differentiation can also be used.