Implicit Differentiation

Differentiate both sides, then solve

When $y$ cannot be isolated (or it is ugly to do so), differentiate every term with respect to $x$, treating $y$ as a function of $x$. Every time $y$ is differentiated, the chain rule contributes a factor $\dfrac{dy}{dx}$.

So $\dfrac{d}{dx}\left(y^3\right)=3y^2\dfrac{dy}{dx}$, and $\dfrac{d}{dx}(xy)=y+x\dfrac{dy}{dx}$ (product rule and chain rule).

Then collect all $\dfrac{dy}{dx}$ terms on one side and factor. The answer normally contains both $x$ and $y$ — that is expected, not a mistake.

For a vertical tangent, look for where the denominator of $\frac{dy}{dx}$ is zero while the numerator is not; for a horizontal tangent, the numerator zero while the denominator is not.