How is the differential definition of money supply, velocity, and GDP defined?
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:
$$ \ln M + \ln V = \ln Y + \ln P \implies \Delta M + \Delta V = \Delta Y + \Delta P$$