Who this is for: Cambridge IGCSE Mathematics (0580/0607) Extended students who want differentiation — finding gradients, stationary points and rates of change — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise differentiation in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the differentiation revision-guide angle, while Tutopiya’s Differentiation subtopic page owns the learning resource and the free differentiation quiz owns the practice.

Differentiation is an Extended-tier topic in Cambridge IGCSE Mathematics (0580/0607). You use it to find the gradient of a curve at a point, locate stationary points, and interpret rates of change in context. Examiners reward correct power-rule application and clear statements about maxima and minima. This guide explains the subtopic, the command words on papers, and where to practise.

Key takeaways

• The derivative dy/dx gives the gradient of the curve y = f(x) at any point.
• Power rule: if y = xⁿ, then dy/dx = nxⁿ⁻¹ (Extended also covers simple multiples).
• Stationary points occur where dy/dx = 0 — then use the second derivative or a sketch to classify max/min.
• In context questions, rate of change means differentiate with respect to time.

What is differentiation in Cambridge IGCSE Maths?

Differentiation is the process of finding the rate at which one quantity changes with respect to another. For a curve y = f(x), dy/dx measures the gradient at each point. In Cambridge IGCSE Mathematics (Extended) you differentiate polynomials using the power rule, find gradients at given x-values, locate stationary points, and solve simple optimisation problems.

Study the notes on Tutopiya’s Differentiation subtopic page before attempting questions.

The core ideas you must master

Idea	What it means	How the exam uses it
dy/dx	Gradient function	”Find dy/dx when y = 3x² − 5x + 2”
Gradient at a point	Substitute x into dy/dx	”Find the gradient of the curve at x = 2”
Stationary point	Where dy/dx = 0	”Find the coordinates of the stationary point”
Maximum / minimum	Second derivative test or sketch	”Determine whether the stationary point is a maximum or minimum”
Rate of change	Derivative in context	”Find the rate at which the area is increasing”

How to differentiate and find a gradient — step by step

1. Write y in index form (e.g. 1/x² = x⁻²).
2. Differentiate term by term using the power rule: multiply by the power, reduce the power by 1.
3. Simplify dy/dx.
4. To find the gradient at x = a, substitute x = a into dy/dx.
5. For stationary points, set dy/dx = 0 and solve for x, then find y from the original equation.

Check your method with the free Differentiation quiz.

Stationary points: maximum or minimum?

Test	Condition	Conclusion
Second derivative d²y/dx² > 0	At stationary x	Minimum
Second derivative d²y/dx² < 0	At stationary x	Maximum
d²y/dx² = 0	Inconclusive	Use sketch or first-derivative sign change

Differentiation in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Find dy/dx	Differentiate y with respect to x	”Find dy/dx when y = x³ − 4x.”
Find the gradient of the curve at the point where x = …	Substitute into dy/dx	”Find the gradient at x = −1.”
Find the coordinates of the stationary point	dy/dx = 0, then find y	”Find the stationary point on y = x² − 6x + 5.”
Show that	Prove a given derivative	”Show that dy/dx = 6x − 4 when y = 3x² − 4x + 1.”
Find the rate of change	Differentiate in context	”The radius increases at … Find dA/dr.”

Worked exam-style stems (how to answer the wording)

1. “Find dy/dx when y = 2x³ − 5x² + 4.” dy/dx = 6x² − 10x. Reward: each term differentiated correctly.
2. “The curve y = x² − 4x + 7 has a stationary point. Find its coordinates.” dy/dx = 2x − 4 = 0 → x = 2, y = 3 → (2, 3). Reward: derivative, solving, y-value.
3. “Find the gradient of y = 1/x at x = 2.” y = x⁻¹ → dy/dx = −x⁻² = −1/x² → at x = 2, gradient = −1/4. Reward: index form, substitution.
4. “Determine whether the stationary point on y = −x² + 4x − 1 is a maximum or a minimum.” d²y/dx² = −2 < 0 → maximum. Reward: second derivative calculated and conclusion stated.
5. “A rectangle has area A = x(10 − 2x). Find the value of x that gives the maximum area.” Expand, differentiate, set dy/dx = 0 → x = 2.5. Reward: forming the function, differentiation, solving — a typical applied stem on Paper 4.

When you can recognise the wording instantly, work the full set on the Algebra topical past-paper questions and the Differentiation quiz.

How differentiation connects to the wider syllabus

Differentiation links to Quadratic Functions for turning points and to graph interpretation across Functions. The Cambridge IGCSE Maths resource hub connects Algebra and Functions revision.

Common mistakes students make

• Forgetting to reduce the power by 1 after multiplying by the index.
• Not rewriting 1/x or √x in index form before differentiating.
• Finding x for a stationary point but forgetting to calculate y.
• Confusing gradient = 0 (stationary) with gradient undefined.
• Classifying max/min without a second derivative or sketch when marks require it.

When you need more support

If differentiation questions keep slipping, retake the Differentiation quiz, work through the Algebra topical past-paper questions, and ask a Cambridge IGCSE Maths tutor for targeted help.

Frequently asked questions

Is differentiation on Core or Extended? Differentiation is an Extended-tier topic in Cambridge IGCSE Mathematics (0580/0607). Core candidates are not examined on it.

What is the power rule? If y = xⁿ, then dy/dx = nxⁿ⁻¹. Differentiate each term separately in a polynomial.

How do I find a stationary point? Set dy/dx = 0, solve for x, substitute back into y to find coordinates.

How do I revise differentiation effectively? Practise differentiating, finding gradients at points, then stationary points. Use the differentiation quiz after each skill.

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