How do you find the derivative of #f(x) = 3 arcsin(x^2)#?

1 Answer
Feb 20, 2017

#dy/dx=(6x)/sqrt(1-x^4)#

Explanation:

Let #y=f(x)#

#y=3arcsin(x^2)#

This suggests that #sin(y/3)=x^2#

Taking the derivative of both sides, we get:

#1/3cos(y/3)dy/dx=2x#

Rearranging and cleaning it up, we get:

#dy/dx=(6x)/(cos(y/3))#

We now need to rewrite #cos(y/3)# in terms of #x#. We can do this using #sin^2A+cos^2A=1#

#cos^2(y/3)+sin^2(y/3)=1# where #sin^2(y/3)=x^4#

#cos^2(y/3)=1-x^4#

#cos(y/3)=sqrt(1-x^4)#

#dy/dx=(6x)/sqrt(1-x^4)#