How is the differential definition of money supply, velocity, and GDP defined? I was reading: This article on money velocity (see the summary section)
Where two formulas are presented:
first an equality:
$$\text{Quantity of Money} (M) \times \text{Velocity of Money} (V)  =  \text{Real GDP} (Y) \times \text{Prices} (P)  $$
or just stated as
$$ MV = YP$$
then its derivative is stated in the article as:
$$ \Delta M + \Delta V =  \Delta Y + \Delta P $$
This second statement makes no sense to me, since if I apply the product rule I should get
$$ M \Delta V + V\Delta M = Y \Delta P + P \Delta Y$$
And I see no way to easily drop the non differential terms. Where did I go wrong here?
That is not a differential form of the equation. $\Delta$ there stands just for changes not differential. You can derive the equation in such form by log-linearization.
Take the original equation:
$$MV=YP$$
Apply natural log to both sides:
$$ \ln M + \ln V = \ln Y + \ln P$$
And now you are done, since in an equation that is in log-log form you can interpret everything in terms of changes.
Hence:
$$ \ln M + \ln V = \ln Y + \ln P \implies \Delta M + \Delta V = \Delta Y + \Delta P$$