Answer
$$
xy^{a}=k
$$
where $a$ and $k$ are constants.
$$
\begin{aligned}
\frac{d y}{d x} &=-\frac{y}{a x}
\end{aligned}
$$
Work Step by Step
$$
xy^{a}=k
$$
where $a$ and $k$ are constants.
Now, we can calculate $\frac{dy}{dx}$ by implicit differentiation, use the product rule as follows:
$$
\begin{aligned}
\frac{d}{d x}\left(x y^{a}\right) &=\frac{d}{d x}(k) \\
x \frac{d}{d x}\left(y^{a}\right)+y^{a}(1) &=0 \\
x\left(a y^{a-1} \frac{d y}{d x}\right)+y^{a} &=0 \\
a x y^{a-1} \frac{d y}{d x} &=-y^{a} \\
\frac{d y}{d x} &=-\frac{y^{a}}{a x y^{a-1}} \\
\frac{d y}{d x} &=-\frac{y}{a x} .
\end{aligned}
$$
