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

1 Answer
Steve M
Feb 21, 2017

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)

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