elasticity from inverse demand I was wondering of my thinking here was right:
Given $$ e=\frac{dQ}{dp}*\frac{p}{Q}, $$ where
$ e $ is elasticity, $ dQ/dp $ is first derivative of demand function, $p$ is price and $Q$ is quantity.
With this expression for $e$, could you then state that:
$$ \frac{dp}{dQ} = \frac{p}{(e*Q)} $$
where $ dp/dQ $ expresses first derivative of inverse demand.
Is this correct?
I will denote the demand function by $Q(p)$ and the inverse demand function by $P(q)$. Then
$$
\forall q: Q(P(q)) = q
$$
so for any $h &gt; 0$ and $q$ we have
$$
\begin{align*}
p &amp; := P(q) \\
p_h &amp; := P(q+h) \\
q &amp; = Q(p) \\
q_h &amp; := Q(p_h) = q+h
\end{align*}
$$
From the definition of derivatives
$$
\frac{\text{d} P(q)}{\text{d} q} := \lim_{h \to 0} \frac{p_h - p}{q_h - q}.
$$
If this is a non zero real number then indeed
$$
\frac{1}{\lim_{h \to 0} \frac{p_h - p}{q_h - q}} = \lim_{h \to 0} \frac{q_h - q}{p_h - p} = \frac{\text{d} Q(p)}{\text{d} p}.
$$

There are some technical considerations. We did not prove that
$$
\lim_{h \to 0} p_h - p = 0.
$$
This is true if $P$ is continuous.
We also assumed that at $q$ and $q+h$ both functions are single valued. If there is a price $p$ where $Q(p) = 0$ then $\frac{\text{d} P(q)}{\text{d} q}$ does not exist.