Lets assume there is a pump or another device that gives more Kinetic Energy to a gas or fluid that is passing through it on an horizontal line, and lets assume perfect conditions like, all the power of the device is only used to do work, there is not an amount that is transformed to heat.
I can say that the power of the device is equal to the change of energy, per time, that the fluid will gain. So, $$ P=\frac{dE}{dt} $$
I know this change of energy will only be in form of Kinetic energy, since the velocity of exit will be greater than the velocity when the fluind enters the device. So, $$ P=\frac{d(KE)}{dt} $$
Now, I know how kinetic energy is defined, so,
$$ P=\frac{1}{2}\frac{d(mv^2)}{dt} $$
Now, I know the velocities will change, and also, there is a thing called "mass flow", which is the amount of mass passing an area, per time, so,
$$ P=\frac{1}{2}\cdot\ v^2 \cdot\frac{dm}{dt} +\frac{2}{2}\cdot\ m \cdot v\cdot\frac{dv}{dt} $$ $$ =\frac{1}{2}\cdot\ v^2 \cdot\frac{dm}{dt} +\ m \cdot v\cdot\frac{dv}{dt} $$
I am exploring this mathematically, using differentials and derivatives, because we are talking about "changes". So, does this makes sense? If not, why? Is that because I am mixing a derivative (velocity) with a flux(mass flow), which is defined by inexact differentials?
Is correct to say Power is equal to "$\frac{dE}{dt}$"?
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