# How do you differentiate g(x) = (5x^6 - 4)cos(5x) using the product rule?

Feb 24, 2018

$g ' \left(x\right) = 30 {x}^{5} \cos \left(5 x\right) - 5 \sin \left(5 x\right) \left(5 {x}^{6} - 4\right)$

#### Explanation:

$\text{differentiate using "color(blue)"product/chain rule}$

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

$g ' \left(x\right) = f \left(x\right) h ' \left(x\right) + h \left(x\right) f ' \left(x\right) \leftarrow \textcolor{b l u e}{\text{product rule}}$

$f \left(x\right) = 5 {x}^{6} - 4 \Rightarrow f ' \left(x\right) = 30 {x}^{5}$

$h \left(x\right) = \cos \left(5 x\right)$

$\Rightarrow h ' \left(x\right) = - \sin \left(5 x\right) \times \frac{d}{\mathrm{dx}} \left(5 x\right) = - 5 \sin \left(5 x\right)$

$\Rightarrow g ' \left(x\right) = - 5 \sin \left(5 x\right) \left(5 {x}^{6} - 4\right) + 30 {x}^{5} \cos \left(5 x\right)$