# What is the derivative of this function sin^3(x)cos(x)?

Feb 28, 2017

Derivative is $3 {\sin}^{2} x {\cos}^{2} x - {\sin}^{4} x$

#### Explanation:

We use the product rule and chain rule here.

Product rule states if $f \left(x\right) = g \left(x\right) h \left(x\right)$, then $\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{dg}}{\mathrm{dx}} \times h \left(x\right) + \frac{\mathrm{dh}}{\mathrm{dx}} \times g \left(x\right)$

and according to chain rule if y=f(u(x) then $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}$.

Hence as $y = f \left(x\right) = {\sin}^{3} x \cos x$

$\frac{\mathrm{df}}{\mathrm{dx}} = 3 {\sin}^{2} x \times \cos x \times \cos x + {\sin}^{3} x \times \left(- \sin x\right)$

= $3 {\sin}^{2} x {\cos}^{2} x - {\sin}^{4} x$