Answer
$$2000\sqrt{3}\pi $$
Work Step by Step
Given $$P=R i^{2},\ \ \ \ i= \sin (4 \pi t)$$
and $R=1000$, $i(1/3)= -\dfrac{\sqrt{3}}{2}$
\begin{align*}
\frac{dP}{dt}&= \frac{dP}{di}\frac{di}{dt}\\
&= (2Ri)( 4\pi \cos (4\pi t))
\end{align*}
Hence
\begin{align*}
\frac{dP}{dt}\bigg|_{t=1/3}&= \frac{dP}{di}\frac{di}{dt}\\
&= (2Ri)( 4\pi \cos (4\pi t)) \\
&= \left(-2000\dfrac{\sqrt{3}}{2}\right) (-2\pi )\\
&= 2000\sqrt{3}\pi
\end{align*}
