inverse function

Question

2. The function f(x) = 3x^3 + 2x + 1 is invertable. If g(x) = f^-1(x), find g'(1).

3 The function f(x)=2x^3+3-5 is invertable. If g(x)=f^-1(x), find g'(-5).

Answer 1

f(x) = 3x³ + 2x + 1

f(0) = 1, so f^-1(1) = 0

f'(x) = 9x² + 2, f'(0) = 2

(f^-1(1))' = 1/ f'(0) = 1/2

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f(x) = 2x³ + 3x - 5

f(0) = -5, so f^-1(-5) = 0

f'(x) = 6x² + 3, f'(0) = 3

(f^-1(-5))' = 1/ f'(0) = 1/3

In general, if f(a) = b, and f is a one to one differentiable function, then (f^-1(b))' = 1 / f'(a)