Answer
The proof is below.
Work Step by Step
Using product rule, we find:
$F=\frac{d(mv)}{dt}$
$F = m \frac{dv}{dt}+v\frac{dm}{dt}$
We know that $a=\frac{dv}{dt}$, so it follows:
$F = ma+v\frac{dm}{dt}$
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