Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only

Differentiation is a cornerstone of A-Level Mathematics, allowing us to analyse rates of change and gradients of curves. In this topic, you will extend your knowledge beyond simple explicit functions (like y = f(x)) to functions defined implicitly (where x and y are mixed, e.g., x² + y² = 25) or parametrically (where x and y are expressed in terms of a third variable, e.g., x = t², y = t³). Mastering these techniques is essential for solving problems involving curves that are not easily written as y = f(x), such as circles, ellipses, or complex paths.

For implicit differentiation, you will learn to differentiate each term with respect to x, treating y as a function of x and applying the chain rule (d/dx of y² = 2y dy/dx). This allows you to find dy/dx even when y cannot be isolated. For parametric differentiation, you use the chain rule in the form dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. These methods are powerful tools for finding gradients, tangents, and normals to curves that arise in geometry, physics, and engineering contexts.

This topic builds directly on your knowledge of basic differentiation (powers, exponentials, trig functions) and the chain rule. It is assessed in Edexcel A-Level Paper 1 (Pure) and Paper 2 (Pure), often in multi-step problems that require combining implicit or parametric differentiation with other skills like equation of a tangent or stationary points. A solid grasp here will also prepare you for more advanced topics like differential equations and optimisation.