# How do you differentiate f(x)=x^3 * sin^2x using the product rule?

Jan 23, 2016

$f ' \left(x\right) = {x}^{2} \sin x \left(3 \sin x + 2 x \cos x\right)$

#### Explanation:

The product rule states that

$f ' \left(x\right) = {\sin}^{2} x \frac{d}{\mathrm{dx}} \left[{x}^{3}\right] + {x}^{3} \frac{d}{\mathrm{dx}} \left[{\sin}^{2} x\right]$

Now, find both of those derivatives.

$\frac{d}{\mathrm{dx}} \left[{x}^{3}\right] = 3 {x}^{2}$

The following will require the power rule with chain rule.

$\frac{d}{\mathrm{dx}} \left[{\sin}^{2} x\right] = 2 \sin x \frac{d}{\mathrm{dx}} \left[\sin x\right] = 2 \sin x \cos x$

Plug these back in to the original equation.

$f ' \left(x\right) = 3 {x}^{2} {\sin}^{2} x + 2 {x}^{3} \sin x \cos x$

This can be factored, but not exceptionally helpfully:

$f ' \left(x\right) = {x}^{2} \sin x \left(3 \sin x + 2 x \cos x\right)$