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

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Answer 1 · 2017-02-20T14:06:24

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

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)#