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As wikipedia says about the second law, As Newton's second law is only valid for constant-mass systems, mass can be taken outside the differentiation operator by the constant factor rule in differentiation This is expanded in the article about variable mass systems. You derived the formula $$ F = m\dot v + \dot m v. $$ where $\dot v \equiv ...


So, $F = \frac{dp}{dt} = \frac{d}{dt}(mv)=m\frac{dv}{dt}+v\frac{dm}{dt} = m\frac{dv}{dt} =\frac{dp}{dv} \times \frac{dv}{dt} = \frac{dp}{dt} = F $ nope, no problem. Note that $\frac{dm}{dt}$ is assumed zero because the situation described by the equation requires that the mass is constant, and hence time invariant. This is of course inapplicable if somehow ...

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