Implicit Differentiation

This lesson covers implicit differentiation, a technique for finding dy/dx when the equation relating x and y is not (or cannot easily be) written in the form y = f(x). Many important curves — circles, ellipses, and other higher-degree curves — are most naturally expressed implicitly, and this technique is essential for A-Level Mathematics.

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What Is an Implicit Equation?

An explicit equation expresses y directly in terms of x: for example, y = x² + 3x.

An implicit equation relates x and y without solving for y: for example, x² + y² = 25 or x³ + y³ = 6xy.

For implicit equations, we cannot always isolate y, so we differentiate both sides of the equation with respect to x, treating y as a function of x.

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The Key Rule

When differentiating a term involving y with respect to x, use the chain rule:

d/dx [f(y)] = f'(y) × dy/dx

In particular:

• d/dx(y²) = 2y × dy/dx
• d/dx(y³) = 3y² × dy/dx
• d/dx(yⁿ) = nyⁿ⁻¹ × dy/dx
• d/dx(sin y) = cos y × dy/dx
• d/dx(eʸ) = eʸ × dy/dx

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Basic Examples