Parametric Differentiation

This lesson covers how to find derivatives (gradients) of curves defined by parametric equations. This is a core A-Level topic that combines parametric equations with differentiation. You will learn how to find dy/dx without first converting to Cartesian form, and how to use this to find equations of tangents and normals to parametric curves.

---

The Chain Rule for Parametric Differentiation

If x and y are both functions of a parameter t, then by the chain rule:

dy/dx = (dy/dt) ÷ (dx/dt)

provided dx/dt ≠ 0.

This formula avoids the need to eliminate the parameter — you can differentiate directly in terms of t.

Derivation: Since y is a function of t and t is a function of x:

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

---

Basic Examples

Example 1: Find dy/dx for the curve x = t², y = t³.

dx/dt = 2t
dy/dt = 3t²

dy/dx = (3t²) / (2t) = 3t/2

Example 2: Find dy/dx for x = 3cos θ, y = 3sin θ.

dx/dθ = −3sin θ
dy/dθ = 3cos θ