# How do you differentiate  f(x) = sin^2 x?

Mar 1, 2017

$2 \sin x \cdot \cos x$

#### Explanation:

It is helpful to write this function as
$f \left(x\right) = {\left(\sin x\right)}^{2}$. Notice that both are equivalent.

Next differentiate using the chain rule.

Recall that the chain rule gives us a method to differentiate compositions. It is similar to the power but with an extra step

$\frac{d}{\mathrm{dx}} f \left(g \left(x\right)\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

For this function:
$f \left(x\right) = g {\left(x\right)}^{2}$ and $g \left(x\right) = \sin x$

If:

$\frac{d}{\mathrm{dx}} f \left(x\right) = 2 g {\left(x\right)}^{1}$ by the power rule

$\frac{d}{\mathrm{dx}} \sin x = \cos x$ (memorize this)

Then:
$\frac{d}{\mathrm{dx}} {\left(\sin x\right)}^{2}$$= 2 \sin x \cdot \cos x$