Am I using quantity and price correctly in trying to calculate price elasticity of supply? The problem I am considering involves a firm that is a price-taker and sells its good at price $p$. From the production function, I calculated the cost function and the profit-maximizing level of output. This allowed me to define my quantity supplied in terms of price. I got 

\begin{equation}
q(p) = \frac{p}{2c}
\end{equation} 

where $c$ is a constant. 

I want to calculate price elasticity of supply. 

\begin{equation}
E_s = \frac{dq}{dp}\frac{p}{q}
\end{equation} 

Thus, I got 
\begin{equation}
\frac{dq(p)}{dp} = \frac{1}{2c}
\end{equation} 
Here's where I am confused: 

Can I now just multiply that derivative by $\frac{p}{q}$ to get the price elasticity of supply? My confusion is that $q(p)$ is a function of $p$. So am I supposed to merely multiply by  $\frac{p}{q}$ or $\frac{p}{q(p)}$?
It does not really mater, as you will get the same numerical values eventually. You may choose whatever is more convenient in each particular situation. Convention is, however, to replase $q$ with $q(p)$ to derive a formula for elasticity as a function of the price only.