Parametric Differentiation

This lesson covers differentiation of parametric equations as required by the Edexcel A-Level Mathematics specification (9MA0). You need to be able to find dy/dx for curves defined parametrically and use it to find tangents and normals.

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What Are Parametric Equations?

In parametric form, both x and y are expressed in terms of a third variable (the parameter), usually denoted t or θ.

x = f(t), y = g(t)

Each value of t gives a point (x, y) on the curve.

Example

The parametric equations x = t², y = 2t describe a parabola (equivalent to x = y²/4 or y² = 4x).

t	-2	-1	0	1	2
x = t²	4	1	0	1	4
y = 2t	-4	-2	0	2	4

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The Formula for dy/dx

To find dy/dx for parametric equations, use:

dy/dx = (dy/dt) / (dx/dt)

provided dx/dt ≠ 0.

This follows from the chain rule: dy/dt = (dy/dx) × (dx/dt), so dy/dx = (dy/dt) / (dx/dt).

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Worked Examples

Worked Example 1