Question

Define the function

Answer 1

Given that

$g(x) = x^3 + x$

the inverse $g^{-1}(x)$ is such that

$g\left(g^{-1}(x)\right) = g^{-1}(x)^3 + g^{-1}(x) = x$

or

$g\left(f(x)\right) = f(x)^3 + f(x) = x$

Differentiating both sides using the chain rule gives

$3f(x)^2f'(x) + f'(x) = 1 \\\\ f'(x) \left(3f(x)^2+1\right) = 1 \\\\ f'(x) = \dfrac1{3f(x)^2+1}$

Then the derivative of f at 2 is

$f'(2) = \dfrac1{3f(2)^2+1} = \boxed{\dfrac14}$