Question: prove that sin^-1(x) = 1/(sqrt 1- x^2)
You want to show that the derivative of sin^(-1)(x) is 1/sqrt(1 - x^2).
Let y = sin^-1(x)
Then, x = sin(y)
Now take the derivative of both sides (wrt y)
dx/dy = cos(y)
dx/dy = sqrt(1 - sin^2(y))
dx/dy = sqrt(1 - x^2)
Now reciprocate both sides.
dy/dx = 1/sqrt(1 - x^2)