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The electrical energy is defined as: $$ E(t) = Q(t)V(t)$$

The general definition of power: $$P(t)=\frac{dE(t)}{dt}$$ and becomes $$P(t) = \frac{dQ(t)}{dt}V(t) + \frac{dV(t)}{dt}Q(t) = I(t)V(t) + \frac{dV(t)}{dt}Q(t)$$

My question: electrical power is often written as $P=I(t)V(t)$. Why is this second term neglected?

For example, the grid has an oscillating voltage, so you cannot neglect $dV(t)/dt$. Also, $Q(t)$ is different from zero because otherwise $dQ(t)/dt$ would also be zero (which is not the case), so why is the power of a resistor then given by only the first term?

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You have the logical order backwards. In circuit theory power is primary and then energy is derived from that. So you start with $P=IV$ and then from that you can derive $E=\int P\ dt=\int IV \ dt$. If $V$ is constant then that works out to $QV$ but that is a derived result assuming constant $V$

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