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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