We first need to work the derivative of #csc^-1(x)#

We can start with this:

#y = csc^-1(x) -> csc(y)=x#

Now differentiate both sides implicitly with respect to #x# to get:

#-dy/dxcsc(y)cot(y)=1#

#-> dy/dx = -1/(csc(y)cot(y))#

Now, using:

#sin^2(y)+cos^2(y)=1#

Divide this identity through by #sin^2(y)# to get:

#1+cot^2(y)=csc^2(y) -> cot(y) = sqrt(csc^2(y)-1)#

We can now plug this into our equation for #dy/dx# to get:

#dy/dx = -1/(csc(y)sqrt(csc^2(y)-1)#

Using #csc(y)=x# we may now rewrite #dy/dx# in terms of #x#:

#dy/dx = -1/(xsqrt(x^2-1)#

So now that we have the derivative of #csc^(-1)x# we can now apply the chain rule to obtain the derivative of #y=csc^-1(4x^2) #

#-> dy/dx= -1/((4x^2)sqrt((4x^2)^2-1)).d/dx(4x^2)#

#=-(8x)/((4x^2)sqrt((4x^2)^2-1))#

Now simplify:

#=-4/(xsqrt(16x^4-1)#