Taking the second derivative of a parametric curve

Consider

$$ \begin{align*} \frac{\text{d}^2y}{\text{d}x^2}&=\frac{\text{d}}{\text{d}x}\left(\frac{\text{d}y}{\text{d}x}\right)\\ &=\frac{\text{d}}{\text{d}t}\left(\frac{\text{d}y}{\text{d}x}\right).\frac{\text{d}t}{\text{d}x}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot \frac{1}{\frac{dx}{dt}}\\ \end{align*}$$
where the last equality is as a result of applying the chain rule.

Their justification is that you can use the same process for $\frac{dy}{dx}$ as for $Y$ since you can now consider $Y_2 = g_2(t) = \frac{dy}{dx}(t)$, that is, you once again have a parametric equation in terms of the parameter $t$, and the parametric equation for $x$ stays the same.