# How do you find the derivative of (x^4 - 1)^10 (2x^4 + 3)^7?

Dec 30, 2015

$56 {x}^{3} {\left({x}^{4} - 1\right)}^{10} {\left(2 {x}^{4} + 3\right)}^{6} + 40 {x}^{3} {\left(2 {x}^{4} + 3\right)}^{7} {\left({x}^{4} - 1\right)}^{9}$

#### Explanation:

Use the product rule :
$\frac{d}{\mathrm{dx}} \left(f \left(x\right) \cdot g \left(x\right)\right) = f \left(x\right) \cdot g ' \left(x\right) + g \left(x\right) \cdot f ' \left(x\right)$

and inside this product rule there will be need for use of the power rule :
$\frac{d}{\mathrm{dx}} {\left[u \left(x\right)\right]}^{n} = n \cdot {\left[u \left(x\right)\right]}^{n - 1} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$.

Hence the derivative of the given function with respect to x is :

${\left({x}^{4} - 1\right)}^{10} \cdot 7 {\left(2 {x}^{4} + 3\right)}^{6} \cdot \left(8 {x}^{3}\right) + {\left(2 {x}^{4} + 3\right)}^{7} \cdot 10 {\left({x}^{4} - 1\right)}^{9} \cdot \left(4 {x}^{3}\right)$

$= 56 {x}^{3} {\left({x}^{4} - 1\right)}^{10} {\left(2 {x}^{4} + 3\right)}^{6} + 40 {x}^{3} {\left(2 {x}^{4} + 3\right)}^{7} {\left({x}^{4} - 1\right)}^{9}$.