How do you find the derivative of y=arcsin(x^4)?

1 Answer
Steve M
Dec 13, 2016

dy/dx = (4x^3)/sqrt(1-x^8)

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.

y=arcsin(x^4) <=> siny=x^4

Differentiate Implicitly:

cosydy/dx = 4x^3 ..... [1]

Using the sin"/"cos identity;

sin^2y+cos^2y -= csc^2y
:. (x^4)^2+cos^2y=1
:. cos^2y=1-x^8
:. cosy=sqrt(1-x^8)

Substituting into [1]
:. sqrt(1-x^8)dy/dx=4x^3
:. dy/dx = (4x^3)/sqrt(1-x^8)

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