Differentiation
In this week’s discussion, we proved the following theorem. As an exercise, replicate the proof via the lemmata below (or devise your own proof).
Theorem 7.1
Let $I \subseteq \mathbb{R}$ be an open interval and $f: I \to \mathbb{R}$ be a continuous injection. If $f$ is differentiable at $x \in I$ and $f’(x) \neq 0$, then $f^{-1}: f(I) \to I$ is differentiable at $y = f(x)$ and $(f^{-1})’(y) = f’(x)^{-1}$.
Lemma 7.1.1
$f$ is strictly monotonic.
Lemma 7.1.2
$f^{-1}$ is continuous.