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)#
