Calculus of a Single Variable, Chapter 5, 5.6, Section 5.6, Problem 49

y = 2arccos(x) - 2sqrt(1-x^2)
First, express the radical in exponent form.
y = 2arccos(x)-2(1-x^2)^(1/2)
To take the derivative of this, use the following formulas:
d/dx(arccos(u)) = -1/sqrt(1- u^2) (du)/dx
d/dx(u^n)= n*u^(n-1) *(du)/dx
Applying these formulas, the derivative of the function will be:
(dy)/dx = d/dx[2arccos(x) - 2(1-x^2)^(1/2)]
(dy)/dx = d/dx[2arccos(x)] - d/dx[2(1-x^2)^(1/2)]
(dy)/dx = 2d/dx[arccos(x)] - 2d/dx[(1-x^2)^(1/2)]
(dy)/dx = 2 * (-1/sqrt(1-x^2))*d/dx(x) - 2 * 1/2(1-x^2)^(-1/2) * d/dx(1-x^2)
(dy)/dx =2 * (-1/sqrt(1-x^2))*1-2*1/2(1-x^2)^(-1/2)* (-2x)
(dy)/dx =2 * (-1/sqrt(1-x^2))*1-2*1/2*1/((1-x^2)^(1/2))* (-2x)
(dy)/dx =2 * (-1/sqrt(1-x^2))*1-2*1/2*1/sqrt(1-x^2)* (-2x)
(dy)/dx = -2/sqrt(1-x^2) +(2x)/sqrt(1 -x^2)
(dy)/dx = (2x-2)/sqrt(1-x^2)

Therefore, the derivative of the function is (dy)/dx = (2x-2)/sqrt(1-x^2) .