Second derivate of parametric equations (Intuitive)

Question

let: $$x=f(t)$$ $$y=g(t)$$ hence: $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} $$

How can I derive the formula for the second derivative of a parametric equation?

Answer 1

If $y'_x=\frac{y'_t}{x'_t}$, then $y'_x=y'_t\cdot t'_x$. Using this for second derivative gives $$y''_{x^2} = (y'_x)'_x=\left( \frac{y'_t}{x'_t} \right)_t\cdot t'_x\quad(1)$$

$y''_{x^2}$ will be derivative of fraction i.e. $$y''_{x^2}=\frac{\frac{dx}{dt}\cdot\frac{d^2y}{dt^2} - \frac{d^2x}{dt^2}\cdot\frac{dy}{dt}}{\left(\frac{dx}{dt}\right)^2}\cdot \frac{dt}{dx}=\\=\frac{x'_ty''_{tt}-y'_tx''_{tt}}{x'^3_t}\quad(2)$$