Quick review of the inverse function theorem:
By definition of inverse function,
f(f⁻¹(x)) = x
Differentiating both sides with respect to x using the chain rule gives
f'(f⁻¹(x)) • (f⁻¹)'(x) = 1
so that the derivative of the inverse is
(f⁻¹)'(x) = 1 / f'(f⁻¹(x))
Now if f(a) = b and a = f⁻¹(b), then the theorem says
(f⁻¹)'(b) = 1 / f'(a)
so that
g'(3) = 1 / f'(1)
We have
f(x) = x³ + x² + x ⇒ f'(x) = 3x² + 2x + 1 ⇒ f'(1) = 3 + 2 + 1 = 6
and so
g'(3) = 1/6
