Implicit Differentiation

This lesson covers implicit differentiation — differentiating equations that are not written in the form y = f(x) — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to be able to find dy/dx for implicitly defined curves and use it to find tangents and normals.

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What Is Implicit Differentiation?

An explicit equation expresses y directly as a function of x:

• y = x² + 3x (explicit)

An implicit equation relates x and y without isolating y:

• x² + y² = 25 (implicit — this is a circle)
• x³ + y³ = 6xy (implicit)
• eˣ + eʸ = x + y (implicit)

For implicit equations, we differentiate both sides with respect to x, treating y as a function of x and applying the chain rule whenever we differentiate a term involving y.

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The Key Rule

When differentiating a function of y with respect to x:

d/dx[f(y)] = f'(y) × dy/dx

This is just the chain rule: y is a function of x, so differentiating f(y) with respect to x requires multiplying by dy/dx.