How can the alternative definition of power, $P = \frac{w}{\Delta t} = F \frac{d}{\Delta t} $, can be derived by substituting the definitions of work and speed into the standard definition of power, $P =\frac{W}{\Delta t}$.

Question

How can the alternative definition of power,  $P = \frac{w}{\Delta t} = F \frac{d}{\Delta t} $, can be derived by substituting the definitions of work and speed into the standard definition of power, $P =\frac{W}{\Delta t}$.

irishboy123 · Answer

power is the rate of change of work with time.  that is P = dW/dt

if the work is a force moving over a distance, then you can say $$dW = F dx,\space\  dP = \frac{d ( F dx)}{ dt} = \frac{dF}{dt} dx + F \frac{dx}{dt}$$.

it all depends on what you are trying to achieve, really.  you can measure energy conversion, say, the movement of a particle, as power.  thus $$P = \frac{ d(\frac{1}{2} m v^{2} )} {dt} = \frac{1}{2} \frac{dm}{dt} v^{2} +  mv \frac{dv}{dt} $$ 
[although dm/dt might have an unusual meaning.]