Explanation of differential equation for Price Elasticity of Demand
Consider a demand equation,
$$Q=\alpha\ +\beta\ P$$

If you were to just use,

$$e_{{\langle p\rangle }} = {\frac  {{\mathrm  {d}}Q}{{\mathrm  {d}}P}}$$

Then the result is only the differentiation with respect to $P$. This doesn't give you the information that you are looking for because you haven't specified the point at which you are measuring elasticity. To do this the second portion of the equation is necessary. Suppose you want to know the elasticity at $P=\$8$. By plugging $\$8$ into the demand equation you can calculate the quantity demanded at $P=\$8$. 

Personally, I have to think of this in two parts, (1) the differentiation with respect to $P$ (2) multiplied by the "spot" of the demand curve that I am examining, ${\frac PQ}$.

$e_{{\langle p\rangle }}={\frac  {{\mathrm  {d}}Q}{{\mathrm  {d}}P}}* {\frac PQ}$

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The number that you calculate is the percent change of a 1% increase in price. For example, if $e=-0.3$, you can say that a 1% increase in price should cause a 0.3% drop in the quantity demanded.