# How do you find the derivative of  arcsin(x^2)?

Feb 21, 2017

$\frac{d}{\mathrm{dx}} \arcsin \left({x}^{2}\right) = \frac{2 x}{\sqrt{1 - {x}^{4}}}$

#### Explanation:

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you can use the chain rule.

Let $y = \arcsin \left({x}^{2}\right) \iff \sin y = {x}^{2}$

Differentiate Implicitly:

$\cos y \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x$ ..... [1]

Using the $\sin \text{/} \cos$ identity;

${\sin}^{2} y + {\cos}^{2} y \equiv 1$
$\therefore {\left({x}^{2}\right)}^{2} + {\cos}^{2} y = 1$
$\therefore {\cos}^{2} y = 1 - {x}^{4}$
$\therefore \cos y = \sqrt{1 - {x}^{4}}$

Substituting into [1]
$\therefore \sqrt{1 - {x}^{4}} \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x}{\sqrt{1 - {x}^{4}}}$