A-Level — mathsAdvanced1 min read

Implicit Differentiation

Differentiate implicitly defined functions at A-Level. Handle equations not in the form y = f(x).

Tutor AI Team2026-02-25T01:57:10.944Z

Implicit differentiation handles equations where $y$ is not explicitly expressed as a function of $x$ , such as $x^2 + y^2 = 25$ .

The Method

Differentiate every term with respect to $x$ . When differentiating a $y$ -term, multiply by $\frac{dy}{dx}$ (chain rule).

Problem

$x^2 + y^2 = 25$ .

$2x + 2y\frac{dy}{dx} = 0$ → $\frac{dy}{dx} = -\frac{x}{y}$ .

Solution

Problem

$x^3 + 3xy + y^3 = 6$ .

$3x^2 + 3y + 3x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0$

$\frac{dy}{dx}(3x + 3y^2) = -3x^2 - 3y$

$\frac{dy}{dx} = \frac{-(x^2 + y)}{x + y^2}$

Solution

Problem

$e^{xy} = x + y$ .

$e^{xy}(y + x\frac{dy}{dx}) = 1 + \frac{dy}{dx}$

Solve for $\frac{dy}{dx}$ .

Solution

Want to check your answers and get step-by-step solutions?

✓
Differentiate every term with respect to $x$ .
✓
$y$ -terms need the chain rule: $\frac{d}{dx}[f(y)] = f'(y)\frac{dy}{dx}$ .
✓
Collect $\frac{dy}{dx}$ terms and solve.

Tags:

#a-level#maths#pure#calculus#implicit

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