Introduction

Sometimes the equation of a curve is not be given in Cartesian form $y=f\left(x\right)$ but in parametric form: $x=h\left(t\right),y=g\left(t\right)$ . 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 $\left(x\left(t\right),y\left(t\right)\right)$ 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