Higher Maths · Relationships and Calculus
Further Differentiation | Higher Maths
Further Differentiation extends basic calculus to include the chain rule and differentiating trigonometric, exponential and logarithmic functions. Expect at least one Paper 2 question worth 4–6 marks on these techniques.
Standard derivatives
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (ex) = ex
d/dx (ln x) = 1/x
Chain rule
If y = f(g(x)), then dy/dx = f′(g(x)) · g′(x).
For y = (something)n: dy/dx = n(something)n−1 × d/dx(something).
Worked example
Worked example — Chain rule application
Problem: Differentiate y = (2x2 + 3)5.
- Let u = 2x2 + 3, so y = u5.du/dx = 4x ; dy/du = 5u4
- Apply chain rule.dy/dx = dy/du × du/dx = 5u4 × 4x
- Substitute u back.dy/dx = 20x(2x2 + 3)4
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Differentiate y = sin 3x.
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dy/dx = 3 cos 3x - Differentiate y = cos(x2).
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dy/dx = −2x sin(x2) - Differentiate y = e2x+1.
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dy/dx = 2e2x+1 - Differentiate y = (3x − 1)4.
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dy/dx = 12(3x − 1)3 - Differentiate y = ln(2x + 5).
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dy/dx = 2/(2x + 5)
Common mistakes
Common mistakes & how to avoid them
- Forgetting to multiply by the derivative of the inside function (the × g′(x) part of the chain rule).
- Differentiating cos x as +sin x — the negative sign is essential.
- Trying to differentiate ln(x2) as 1/x2. Use chain rule: derivative is 2x/x2 = 2/x.
Frequently asked questions
Are the standard derivatives on the formula sheet?
Yes — derivatives of sin, cos, ex and ln x are all on the SQA Higher formula sheet. The chain rule is not formally listed but you must know how to apply it.
Do I need product or quotient rule for Higher?
No — these are reserved for Advanced Higher. Higher only uses the chain rule plus the basic derivative rules.
Is "Further Differentiation" tested separately?
It typically appears as a single multi-part Paper 2 question or as part of a larger optimisation problem.
Related Higher Maths topics
These topics often appear together in SQA exam questions.
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