# What is the derivative of (sin(pix))^2?

Mar 23, 2017

$2 \pi \sin \pi x \cos \pi x = \pi \sin 2 \pi x$

#### Explanation:

Use the power rule $\left({u}^{n}\right) ' = n {u}^{n - 1} \cdot u '$ and
the chain rule $\left(\sin v\right) ' = v ' \cos v$

Let $u = \sin \left(\pi x\right)$, $v = \pi x$, $v ' = \pi$

So $u ' = \left(\sin \pi x\right) ' = \pi \cos \pi x$

$\left({\left(\sin \pi x\right)}^{2}\right) ' = 2 \left(\sin \left(\pi x\right)\right) \left(\pi \cos \pi x\right) = 2 \pi \sin \pi x \cos \pi x$

Use the Double Angle Formula $\sin 2 u = 2 \sin u \cos u$:

$2 \pi \sin \pi x \cos \pi x = \pi \sin 2 \pi x$