How do I compute the derivative of this inverse function?

Question

Let $$f(x)=\frac{1}{16}(e^{\arctan(\frac{x}{7})} + \frac{x}{7})$$ You are given that $f$ is a one-to-one function and its inverse function $f^{-1}$ is a differentiable function on $\mathbb{R}$. Also $f(0)=\frac{1}{16}$. What is the value of $(f^{-1})'(1/16)$?

Answer 1

We have

$$(f^{-1})'(\frac{1}{16})= \frac{1}{f'(f^{-1}(\frac{1}{16})}= \frac{1}{f'(0)}.$$

Answer 2

If $g$ is the inverse of $f$ then:

$f'(x) \cdot g'(x) = 1$ for all $x$

So just do these steps:

1. find $f'$
2. find $f'(1/16)$
3. find $g'(1/16) = 1 / f'(1/16)$