Relative velocity of exhaust in Ideal Rocket Equation

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In this derivation of the Ideal Rocket Equation (https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation#Most_popular_derivation), they use the fact that $V_e = V - v_e$, or that the velocity of the exhaust in the observer frame $V_e$ is the velocity of the rocket $V$ minus the speed of the exhaust relative to the rocket $v_e$.

But since the rocket has sped up by $dV$ during the time it took to eject the exhaust, shouldn't this expression really be $V_e = V + dV - v_e$? Note that even if my reasoning is correct, it doesn't actually make a difference in the derivation because dV ends up being in a product of two differentials, which equals 0.

 

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Note that even if my reasoning is correct, it doesn't actually make a difference in the derivation because dV ends up being in a product of two differentials, which equals 0.

Exactly. Two ways to derive the same result. Both are fine.