What is the derivative of arcsin(x^4)?

1 Answer
Jun 2, 2017

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

Explanation:

Let u=x^4 then (du)/(dx)=4x^3

Let y=arcsin(u)

Take the arcsin of both sides

sin(y)=u

Differentiate both sides with respect to u

dy/(du)cos(y)=1

Divide both sides by cos(y)

dy/(du)=1/cos(y)

recall that Sin^2(y)+cos^2(y)=1

So cos(y)=sqrt(1-sin^2(y))
NOTE: Take only the positive root

From above we said that sin(y)=u so we can write

cos(y)=sqrt(1-u^2)

So dy/(du)=1/sqrt(1-u^2)

Recall that u=x^4

So dy/(du)=1/sqrt(1-(x^4)^2)

Now using the chain rule

dy/dx=dy/(du)(du)/(dx)

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

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