How do you find the second derivative of a parametric function?

Oct 2, 2014

Let $\left\{\begin{matrix}x = x \left(t\right) \\ y = y \left(t\right)\end{matrix}\right.$.

First Derivative

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}} = \frac{y ' \left(t\right)}{x ' \left(t\right)}$

Second Derivative

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{d}{\mathrm{dx}} \left[\frac{y ' \left(t\right)}{x ' \left(t\right)}\right] = \frac{1}{\frac{\mathrm{dx}}{\mathrm{dt}}} \frac{d}{\mathrm{dt}} \left[\frac{y ' \left(t\right)}{x ' \left(t\right)}\right]$

$= \frac{1}{x ' \left(t\right)} \cdot \frac{y ' ' \left(t\right) x ' \left(t\right) - y ' \left(t\right) x ' ' \left(t\right)}{{\left[x ' \left(t\right)\right]}^{2}}$

$= \frac{y ' ' \left(t\right) x ' \left(t\right) - y ' \left(t\right) x ' ' \left(t\right)}{{\left[x ' \left(t\right)\right]}^{3}}$

I hope that this was helpful.