Question

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

Answer 1

`d/dx arcsin(x^2)= (2x)/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(x^2) <=> siny=x^2`

Differentiate Implicitly:

`cosydy/dx = 2x` ..... [1]

Using the `sin"/"cos` identity;

`sin^2y+cos^2y -= 1`
`:. (x^2)^2+cos^2y=1`
`:. cos^2y=1-x^4`
`:. cosy=sqrt(1-x^4)`

Substituting into [1]
`:. sqrt(1-x^4)dy/dx=2x`
`:. dy/dx = (2x)/sqrt(1-x^4)`