Math Question?

Question

For –л/2≤x≤ л/2, sin x is a 1-1 function. Define f(x)= sinx on [-л/2, л/2]. Show that

(f-1)’(x)= 1/√(1-x²)

Answer 1

Use the chain rule and the fact sin(f^{-1}(x)) = x. Differentiating both sides:

cos(f^{-1}(x))*(f^{-1})'(x) = 1.

cos(sin^{-1}(x)) = sqrt(1-x^2), so you get the formula.