Implicit and parametric gradients

The idea

Implicit: differentiate every term with respect to $x$, and multiply each $y$ term by $\frac{dy}{dx}$ (chain rule). Parametric: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

1. Implicit: differentiate term by term; $y^2 \to 2y\frac{dy}{dx}$, $xy \to y + x\frac{dy}{dx}$.
2. Collect the $\frac{dy}{dx}$ terms and factorise.
3. Parametric: find $\frac{dx}{dt}$ and $\frac{dy}{dt}$, then divide.

Watch out $y$ depends on $x$: $\frac{d}{dx}(y^2)$ is $2y\frac{dy}{dx}$, not $2y$.