How to properly calculate current of a variable capacitor?

Question

So the formula i(t)=CdV/dt works for a constant value of a capacitor, but in a variable capacitor, i found the next formula on another site

i(t)=C(t)*dV(t)/dt + V(t)*dC(t)/dt

Is this formula correct?

How do you get the new formula from the previous one, with implicit differentiation?

In which specific book can i find this information?

Thanks for the future answers.

Answer 1

Use the product rule to differentiate on \$Q(t)=C(t)V(t)\$:

\$\dfrac{d}{dt} f(t)g(t) = \dfrac{df(t)}{dt}g(t) + f(t)\dfrac{dg(t)}{dt}\$