We first need to work the derivative of csc^-1(x)csc−1(x)
We can start with this:
y = csc^-1(x) -> csc(y)=xy=csc−1(x)→csc(y)=x
Now differentiate both sides implicitly with respect to xx to get:
-dy/dxcsc(y)cot(y)=1−dydxcsc(y)cot(y)=1
-> dy/dx = -1/(csc(y)cot(y))→dydx=−1csc(y)cot(y)
Now, using:
sin^2(y)+cos^2(y)=1sin2(y)+cos2(y)=1
Divide this identity through by sin^2(y)sin2(y) to get:
1+cot^2(y)=csc^2(y) -> cot(y) = sqrt(csc^2(y)-1)1+cot2(y)=csc2(y)→cot(y)=√csc2(y)−1
We can now plug this into our equation for dy/dxdydx to get:
dy/dx = -1/(csc(y)sqrt(csc^2(y)-1)dydx=−1csc(y)√csc2(y)−1
Using csc(y)=xcsc(y)=x we may now rewrite dy/dxdydx in terms of xx:
dy/dx = -1/(xsqrt(x^2-1)dydx=−1x√x2−1
So now that we have the derivative of csc^(-1)xcsc−1x we can now apply the chain rule to obtain the derivative of y=csc^-1(4x^2) y=csc−1(4x2)
-> dy/dx= -1/((4x^2)sqrt((4x^2)^2-1)).d/dx(4x^2)→dydx=−1(4x2)√(4x2)2−1.ddx(4x2)
=-(8x)/((4x^2)sqrt((4x^2)^2-1))=−8x(4x2)√(4x2)2−1
Now simplify:
=-4/(xsqrt(16x^4-1)=−4x√16x4−1
