A-level · Beta

Product and quotient rules

Compare the separate rates with the derivative of a product or quotient.

Choose two functions and move the trace point. Compare both contributions to the derivative.

(fg)′=f′g+fg′(fg)\prime=f\prime g+fg\prime
Choose two functions and move the trace point. Compare both contributions to the derivative.-2-15-1-7.50017.5215xy

At x = 1, the gradient is 10.

f = ax + b
3
g = xⁿ + c
2
f′
2
g′
2
Function value
6
Tangent gradient
10

The blue graph is the combined function; green is its tangent. The graph window clips values outside ±15 and breaks at singularities. Sine uses radians. Differentiating a product does not mean multiplying derivatives.

Predict, test and explain

Step 1 of 3

Predict

Is (fg)′ equal to f′g′?

InvestigateCan you differentiate a product by multiplying the two derivatives?

Explore and compare

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

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

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Is (fg)′ equal to f′g′?

(fg)′=f′g+fg′(fg)\prime=f\prime g+fg\prime

Hint

Differentiate each factor first, then assemble the complete rule.

Reveal answer

Both factors contribute to the change. For a quotient, the denominator must be nonzero.

1010

Build an explanation

Hint 1

Differentiate each factor first, then assemble the complete rule.

Hint 2

Both factors contribute to the change. For a quotient, the denominator must be nonzero.

Hint 3

Use the model to check a prediction, then explain the result.

Worked example

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

Find the derivative of (2x + 1)(x² + 1) at x = 1.

  1. f = 3 and f′ = 2 at x = 1.
  2. g = 2 and g′ = 2 at x = 1.
  3. f′g + fg′ = 2 × 2 + 3 × 2.

Answer: Gradient 10.

Connect this idea

Watch out: Both factors contribute to the change. For a quotient, the denominator must be nonzero.

Specification and learning route

AQA G4 · Edexcel Pure 7.4

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

Useful starting knowledge: Basic derivatives; Products and algebraic fractions; Function notation.

AQA specification · Edexcel specification

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