# Derivative of Parametric Functions

## Key Questions

• 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.

• To find the derivative of a parametric function, you use the formula:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}$, which is a rearranged form of the chain rule.

To use this, we must first derive $y$ and $x$ separately, then place the result of $\frac{\mathrm{dy}}{\mathrm{dt}}$over $\frac{\mathrm{dx}}{\mathrm{dt}}$.

$y = {t}^{2} + 2$

$\frac{\mathrm{dy}}{\mathrm{dt}} = 2 t$ (Power Rule)

$x = t \sin \left(t\right)$

$\frac{\mathrm{dx}}{\mathrm{dt}} = \sin \left(t\right) + t \cos \left(t\right)$ (Product Rule)

Placing these into our formula for the derivative of parametric equations, we have:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}} = \frac{2 t}{\sin \left(t\right) + t \cos \left(t\right)}$

• For the parametric equations

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

we can find the derivative

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