Q.If $x = a t^2$ and $y = 2 a t$, find $\dfrac{dy}{dx}$ in terms of $t$.

Both $x$ and $y$ are given in terms of the parameter $t$, so use $\tfrac{dy}{dx}=\tfrac{dy/dt}{dx/dt}$.

The curve is given parametrically, so apply the parametric rule (§7), $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.

Differentiate each with respect to $t$.

Divide.

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2a}{2at} = \frac{1}{t}.$ …

The parameter can be eliminated to give $y^2=4ax$ (a parabola) and then differentiated implicitly, as in the check — but differentiating the two parametric equations sepa …

A common mistake is inverting the ratio to $\tfrac{dx/dt}{dy/dt}=t$ (which is $\tfrac{dx}{dy}$, not $\tfrac{dy}{dx}$), or attempting $\tfrac{y}{x}$ directly without differentiating. Also, leaving the answer in terms of …