(Solved): The power \( P \) in a circuit is \( P=R i^{2} \), where \( R \) is resistance and \( i \) is the c ...

The power \( P \) in a circuit is \( P=R i^{2} \), where \( R \) is resistance and \( i \) is the current. Find \( \frac{d P}{d t} \) at \( t=\frac{1}{3} \) if \( R=1000 \Omega \) and \( i \) varies according to \( i=\sin (4 \pi t) \) (time in seconds). (Use symbolic notation and fractions where needed.) \( \left.\frac{d P}{d t}\right|_{t=\frac{1}{1}} \)