Question

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

Answer 1

`2 pi sin pi x cos pi x = pi sin 2pix`

Explanation:

Use the power rule `(u^n)' = n u^(n-1)*u'` and
the chain rule `(sin v)' = v' cos v`

Let `u = sin (pi x)`, `v = pi x`, `v' = pi`

So `u' = (sin pi x)' = pi cos pi x`

`((sin pi x)^2)' = 2(sin (pix)) (pi cos pi x) = 2 pi sin pi x cos pi x`

Use the Double Angle Formula `sin 2u = 2 sin u cos u`:

`2 pi sin pi x cos pi x = pi sin 2pix`