Average Power and Instantaneous Power

I am confused between the usage of the terms average power and instantaneous power. What is the main difference between the two?

Answer

The amount of work performed during "a period of time"

$\Rightarrow$ Average Power $= \frac{\Delta W}{\Delta t}$

for example, the work is $W_1=3 \, \mathrm{ J}$ at time $t_1= 2\, \mathrm{ sec.}$ and $W_2=7 \, \mathrm{ J}$ at time $t_2= 13\, \mathrm{ sec.}$

$\Rightarrow$ the duration is $\Delta t = t_2-t_1=13-2\, (\mathrm{ sec.})$

$\Rightarrow$ the amount of work is $\Delta W = W_2-W_1=7-3\, (\mathrm{ J})$

$\Rightarrow$ Average Power is $\frac{\Delta W}{\Delta t}=\frac{7-3}{13-2}=\frac{4}{11}\, (\mathrm{ J/s })$

If the time interval $\Delta t \to 0 \Rightarrow \Delta t=dt, \Delta W=dW$

(It means that the Power is at some moment. )

$\Rightarrow$ Instantaneous Power $= \frac{d W}{d t}$