Let f be differentiable on an interval (a,b) and assume that f has an inverse f^-1 that is also differentiable on the interval (f(a), f(b)). (a) Show that (f^-1) (xf’(f^1(x) = 1 for all 2 € (f(a), f(b)). [3] (b) Derive from the previous question that (f^-1)(x)= 1/(f’(f^-1(x)) for all x € (f(a), f(b)). [1] (c) Determine the derivative off where f(x) = sin^1(√1-x^2) [8]
5.2 Apply the mean value theorem to the function f(t) = ln(t) on the interval [1, 1 + x)(x > 0) to show that for all
x > 0, 1/(1+x) < ln (1+x) < x [8]