Higher Maths · Relationships and Calculus
Differentiation | Higher Maths
Differentiation is one of the highest-mark topics on Higher Maths. Almost every Paper 2 has an extended differentiation question worth 8–10 marks. You must be confident with the basic rules, finding stationary points and writing the equation of a tangent line.
Basic differentiation
Power rule: if y = xn then dy/dx = nxn−1
Constant: d/dx (c) = 0
Sum: d/dx (f + g) = f′ + g′
Constant multiple: d/dx (kf) = k · f′
Stationary points: solve dy/dx = 0
Nature of stationary point: examine sign of dy/dx either side, or use the second derivative d2y/dx2.
Worked example
Worked example — Stationary points
Problem: Find the stationary points of the function y = x3 − 6x2 + 9x + 1 and determine their nature.
- Differentiate.dy/dx = 3x2 − 12x + 9
- Set dy/dx = 0 and solve.3x2 − 12x + 9 = 0 ⇒ 3(x − 1)(x − 3) = 0 ⇒ x = 1 or x = 3
- Find y at each stationary point.When x = 1: y = 5. When x = 3: y = 1. So stationary points are (1, 5) and (3, 1).
- Determine nature using a sign table for dy/dx.Either side of x = 1: dy/dx changes from + to − ⇒ maximum at (1, 5). Either side of x = 3: dy/dx changes from − to + ⇒ minimum at (3, 1).
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Differentiate y = 4x3 − 7x + 2.
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dy/dx = 12x2 − 7 - Find the equation of the tangent to y = x2 − 3x + 1 at the point where x = 2.
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At x = 2, y = −1, dy/dx = 1. Tangent: y = x − 3 - A function is increasing where dy/dx > 0. For y = x3 − 12x, find the values of x for which the function is increasing.
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dy/dx = 3x2 − 12 > 0 ⇒ x2 > 4 ⇒ x < −2 or x > 2 - Differentiate f(x) = (2x − 1)/√x.
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Rewrite f(x) = 2x1/2 − x−1/2. f′(x) = x−1/2 + (1/2)x−3/2 - A box has volume V(x) = x(20 − 2x)2. Show that V is maximum when x = 10/3 and find the maximum volume.
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dV/dx = (20 − 2x)2 − 4x(20 − 2x) = 0 ⇒ (20 − 2x)(20 − 6x) = 0 ⇒ x = 10 (rejected) or x = 10/3. Maximum volume = 16000/27 cubic units.
Common mistakes
Common mistakes & how to avoid them
- Forgetting to bring the original power down before subtracting 1 — for y = x5, dy/dx = 5x4, not x4.
- Differentiating fractions or surds without first rewriting them with negative or fractional indices.
- Stating "minimum" or "maximum" without supporting it with a sign table or second derivative — SQA marking schemes require justification.
Frequently asked questions
What is differentiation actually used for?
Differentiation gives the rate of change. In Higher you will use it to find gradients of curves, equations of tangents, maximum and minimum values for optimisation, and to analyse where functions are increasing or decreasing.
Do I need to memorise the differentiation rule?
Yes — the power rule (d/dx (xn) = nxn−1) is not on the SQA Higher formula sheet. You must know it from memory.
How many marks is differentiation worth on the Higher exam?
Across both papers, differentiation usually accounts for 12–18 marks out of 100, including stationary points, tangents and optimisation problems.
Related Higher Maths topics
These topics often appear together in SQA exam questions.
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