Problem Let $f$ be a continuous one-one function defined on an interval and suppose that $f$ is differentiable at $f^{-1}(b)$ with the derivative $f'(f^{-1}(x)) \neq 0$. Prove that $f^{-1}(x)$ is also differentiable at $b$.
I read the proof in Spivak Calculus book, but it's quite confusing to me. Then I looked up online, and I saw exact the same proof everywhere. So I wonder is there an alternative proof to this problem that I'm not aware of?
Here is the proof from Spivak book
Theorem 5 (Calculus by Spivak 4th edition, page 237-238)
Proof. Let $b = f(a)$. Then $\displaystyle\lim_{h\to 0} \dfrac{f^{-1}(b + h) - f^{-1}(b)}{h} = \displaystyle\lim_{h\to 0} \dfrac{f^{-1}(b + h) - a}{h}$ Now every number $b + h$ in the domain of $f^{-1}$ can be written in the form $b + h = f(a + k)$ for a unique $k$. Then $\displaystyle\lim_{h\to 0} \dfrac{f^{-1}(b + h) - a}{h} = \displaystyle\lim_{h\to 0}\dfrac{f^{-1}(f(a + k)) - a}{f(a + k) - b} = \displaystyle\lim_{h\to 0} \dfrac{k}{f(a + k) - f(a)}$ We have $f^{-1}(b + h) = a + k \Leftrightarrow k = f^{-1}(b + h) - f^{-1}(b)$ Since $f$ is continuous and one-one, $f^{-1}$ is also continuous at $b$. This means that $k$ approaches $0$ as $h$ approaches $0$. Since $\displaystyle\lim_{k\to 0} \dfrac{f(a + k) - f(a)}{k} = f'(a) = f'(f^{-1}(b)) \neq 0$ this implies that $(f^{-1})'(b) = \dfrac{1}{f'(f^{-1}(b))}$