Answer
The demand equation for a certain product is
$$
2p^{2}+q^{2}=1600;
$$
(a) $\frac{d q}{d p}$: the rate of change of demand with respect to price is given by:
$$
\begin{aligned}
\frac{dq}{d p}&= \frac{-2p}{q }.
\end{aligned}
$$
(b) $\frac{d p}{d q}$: the rate of change of price with respect to demand is given by:
$$
\begin{aligned}
\frac{dp}{d q} &= \frac{-q}{2p} .\\
\end{aligned}
$$
Work Step by Step
The demand equation for a certain product is
$$
2p^{2}+q^{2}=1600;
$$
(a) $\frac{d q}{d p}$: the rate of change of demand with respect to price.
Now, we can calculate $\frac{d q}{d p}$ by implicit differentiation,
$$
\begin{aligned}
\frac{d}{d p}(2p^{2}+q^{2} )&=\frac{d}{d p}(1600 ) \\
(4p+2q \frac{dq}{d p} )&=0\\
2q \frac{dq}{d p} &=-4p\\
\frac{dq}{d p}&= \frac{-4p}{2q }\\
\frac{dq}{d p}&= \frac{-2p}{q }.\\
\end{aligned}
$$
(b) $\frac{d p}{d q}$: the rate of change of price with respect to demand .
Now, we can calculate $\frac{d p}{d q}$ by implicit differentiation,
$$
\begin{aligned}
\frac{d}{d q}(2p^{2}+q^{2} )&=\frac{d}{d q}(1600 ) \\
(4p \frac{dp}{d q}+2q )&=0\\
4p \frac{dp}{d q} &=-2q\\
\frac{dp}{d q} &= \frac{-2q}{4p} \\
\frac{dp}{d q} &= \frac{-q}{2p} .\\
\end{aligned}
$$
