A-level · Beta

Implicit differentiation

Trace a circle or ellipse and connect the chain rule to tangent gradients.

x^2+y^2=9

Scroll the diagram sideways to see every label.

Point P ≈ (2.1213, 2.1213). Tangent gradient: -1.

Differentiate both sides

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

For y ≠ 0, $\frac{dy}{dx}=-\frac xy$ . The factor dy/dx comes from the chain rule.

The green dashed line is the tangent at P.

Both axes use the same scale. The point angle locates P on the circle or parametrised ellipse; it is not generally the geometric polar angle for an ellipse. Coordinates and gradients are rounded to 4 decimal places. The model uses positive radii only.

Predict, test and explain

Step 1 of 3

Predict

When differentiating y² with respect to x, can you write only 2y?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

When differentiating y² with respect to x, can you write only 2y?

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

Hint

Differentiate both sides with respect to x and collect all dy/dx terms.

Reveal answer

Here y depends on x. Differentiate y² using the chain rule, keeping dy/dx.

$\frac{dy}{dx}=-\frac xy$

Build an explanation

Hint 1

Differentiate both sides with respect to x and collect all dy/dx terms.

Hint 2

Here y depends on x. Differentiate y² using the chain rule, keeping dy/dx.

Hint 3

Explain why the rule is valid, including its assumptions.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the tangent to x² + y² = 25 at (3, 4).

Differentiate: 2x + 2y dy/dx = 0.
At (3, 4), dy/dx = −3/4.
Use point-gradient form through (3, 4).

Answer: y − 4 = −3(x − 3)/4.

Connect this idea

Watch out: Here y depends on x. Differentiate y² using the chain rule, keeping dy/dx.

Specification and learning route

AQA G5 · Edexcel Pure 7.5

A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Chain rule; Curve gradients; Circle equations.

AQA specification · Edexcel specification