I'm helping a family friend with an introductory business mathematics course; I have no background in economics, so I've been previewing the applications covered in her textbook, and just read about the concept of elasticity.
The derivation of the formula $E=-\frac{dq}{dp} \cdot \frac{p}{q}$ is straightforward, as are the ideas that when $E<1$, demand reacts weakly to an increase in price, so raising the price will lead to an increase in revenue, and that the opposite will happen when $E>1$. The textbook also mentions* (though only an intuitive explanation is provided) that the price which maximizes revenue is the one that gives unit elasticity (i.e. $E=1$.)
What I'm not quite grasping is what the use of this concept is to begin with. Maximizing revenue would seem to me to just come from $\frac{dR}{dp}=0$, and it's not difficult to see the two are equivalent:
$$R=pq \implies \frac{dR}{dp} = q + p \cdot \frac{dq}{dp}$$
and therefore
$$\frac{dR}{dp} = 0 \iff -p \cdot \frac{dq}{dp} = q \iff E=1$$
and similarly $E<1 \iff \frac{dR}{dp}>0$ and $E>1 \iff \frac{dR}{dp}<0$.
I do understand that $E$ gives us the percent change in demand for each unit percent increase in price, but does that have any practical use other than to maximize revenue? If not, why use $E$ at all rather than just $\frac{dR}{dp}$?
*What the text actually says is that "in ordinary cases the price that maximizes revenue must give unit elasticity." However, I'm unclear on when the derivation I showed above would not work.