Power When Voltage and Current Both Change

Electrical power is voltage times current. When both are changing, the power changes for two reasons at once: because the voltage moved and because the current moved. The product rule adds those two effects.

How it works

Product rule

In one sentence

If $f$ and $g$ are differentiable at $x$ , then $(fg)'(x) = f'(x)g(x) + f(x)g'(x)$ .

General formula

{\htmlData{sym=0}{P(t)}} = {\htmlData{sym=1}{V(t)}}\,{\htmlData{sym=2}{I(t)}} \quad\Rightarrow\quad P'(t) = {\htmlData{sym=3}{V'(t)}}\,{\htmlData{sym=2}{I(t)}} + {\htmlData{sym=1}{V(t)}}\,{\htmlData{sym=4}{I'(t)}}

Picture $V \times I$ as the area of a rectangle. Growing each side adds a thin strip along the other side; the tiny corner where both grow is negligible.

What the symbols mean

$P(t)$
Power
Watts delivered at time t.
$V(t)$
Voltage
Volts across the component.
$I(t)$
Current
Amps through it.
$V'(t)$
Voltage rate
How fast the voltage is changing.
$I'(t)$
Current rate
How fast the current is changing.

Over a short time, $V$ changes by $\Delta V$ and $I$ by $\Delta I$ . The power changes by $(V + \Delta V)(I + \Delta I) - VI$ .
That's $\Delta V\,I + V\,\Delta I + \Delta V\,\Delta I$ ; divide by $\Delta t$ and the last term vanishes in the limit.
So $P' = V'I + VI'$ : the rate from the voltage changing, plus the rate from the current changing.
It is not $V'I'$ : that would miss most of the change.

Learn the math

Product and Quotient RulesCalc I · Lesson 17, Derivatives