Some of the content of this guide was modeled after a guide published by OpenStax and has been adapted for the NWP Learning Commons in March 2022. This work is licensed under a Creative Commons BY 4.0 International License.
Power is the amount of work done in a specific unit of time. Instantaneous power is work done over an instantaneous point in time.
Average power can be found looking at the change in work divided by the change in time.
$${P_{avg}}=\frac{{W_2}-{W_1}}{{t_2}-{t_1}}=\frac{\Delta{W}}{\Delta{t}}$$
As that interval of time reduces down to a limit of zero, the average power becomes instantaneous power.
$${P_{inst}}=P=\lim_{\Delta{t}\to0}\frac{\Delta{W}}{\Delta{t}}=\frac{d{W}}{d{t}}$$
Work can also be defined in terms of power versus time:
$${W}=\int_{t_1}^{t_2}{P\,{dt}}={P}\,{({t_2}-{t_1})}$$