TITLE: Alternative proof for differentiability of inverse function?
QUESTION [6 upvotes]: 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
    $$(f^{-1})'(b) = \dfrac{1}{f'(f^{-1}(b))}$$

REPLY [5 votes]: The strategy of the proof is simple, and the difficulty lies in the existence of the limit for the incremental ration of $f^{-1}$. Indeed, if $f^{-1}$ is differentiable, then $f^{-1} \circ f = \operatorname{Id}$ implies $$Df^{-1}(f(x))Df(x)=1$$ for every $x$ in the domain of $f$, and you conclude.
On the other hand, the existence of the derivative of $f^{-1}$ is a standard application of the change-of-variable rule for limits, which is in turn a corollary of the theorem on limits of composite functions.