How does calculus derive the power formula P = IV from first principles?

[dlacombe13]

I am at the stage of my education where I am seeing calculus (mainly differential equations) popping up in my engineering courses. However, I have just started multi-variable calculus, so I have not taken differential equations yet. My professors basically show us how certain equations are derived, but do not expect us to derive them ourselves yet. However, it bothers me not knowing at least conceptually how these equations even come to be. For example:

$$power = \frac{dU}{dt} = \frac{dU}{dq} * \frac{dq}{dt} = iv$$

I can see that current = energy/charge and that voltage = charge/time, but how exactly is calculus applied to the original power formula to come to the conclusion that power = iv? What is the mathematical process?

 

[BvU]

current = energy/charge and that voltage = charge/time

Actually it's the other way around:

voltage = energy/charge and current = charge/time
​

The mathematical 'process' is the chain rule (in the 'other direction'):

if a is a function of b which in turn is a function of c, so a(b(c)), then $${da\over dc} = {da\over db} \; {db\over dc}$$​