### Introduction

Sometimes the equation of a curve is not be given in Cartesian form $y=f\left(x\right)$ but in parametric form: $x=h\left(t\right),\phantom{\rule{1em}{0ex}}y=g\left(t\right)$ . In this Section we see how to calculate the derivative $\frac{dy}{dx}$ from a knowledge of the so-called parametric derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ . We then extend this to the determination of the second derivative $\frac{{d}^{2}y}{d{x}^{2}}$ .

Parametric functions arise often in particle dynamics in which the parameter $t$ represents the time and $\left(x\left(t\right),\phantom{\rule{1em}{0ex}}y\left(t\right)\right)$ then represents the position of a particle as it varies with time.

#### Prerequisites

• be able to differentiate standard functions
• be able to plot a curve given in parametric form

#### Learning Outcomes

• find first and second derivatives when the equation of a curve is given in parametric form