Question

Define the function

3
g(x) = x^3+ x
. If
f(x) = g^-1(x)
and
f(2) = 1
what is the value of
f'(2)
?

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) = \frac1{3f(x)^2+1}`

Then the derivative of f at 2 is

`f'(2) = \frac1{3f(2)^2+1} = \boxed{\frac14}`