IB Maths AA SL Topic 5 — Calculus Paper 1 & 2 🎯 Skill ~3 min practice

AA SL Differentiating Polynomials skills

Drop the power, multiply by the old one — the single most-used technique in IB calculus. Master this and you’ll handle 80% of Paper 1 differentiation questions on autopilot.

The Method

d/dx[axⁿ] = anxⁿ⁻¹

Bring the power down as a multiplier in front.
Reduce the power by 1.
Constants disappear (their derivative is 0). A bare x becomes 1.

y = 3x⁵

→

5·3x⁴bring 5 down

→

15x⁴simplify

Worked examples

WE 1 EASY

Differentiate y = 5x⁴ − 3x² + 7x − 1

apply the rule term-by-term 5x⁴ → 4·5x³ = 20x³ −3x² → 2·(−3)x = −6x 7x → 7 −1 → 0 (constant) dy/dx = 20x³ − 6x + 7 treat each term separately — sums and differences differentiate term-by-term!

WE 2 MEDIUM

Differentiate y = 8√x + 6/x²

step 1 — rewrite as powers of x y = 8x^(1/2) + 6x⁻²step 2 — apply the rule 8x^(1/2) → ½·8 x^(−1/2) = 4x^(−1/2) 6x⁻² → −2·6 x⁻³ = −12x⁻³step 3 — tidy back into nice form dy/dx = 4/√x − 12/x³ always rewrite roots and fractions as powers BEFORE differentiating!

WE 3 HARD

Differentiate y = (3x³ + 6x²) / (2x)

step 1 — split the fraction first y = 3x³/(2x) + 6x²/(2x) y = (3/2)x² + 3xstep 2 — apply the rule (3/2)x² → 2·(3/2)x = 3x 3x → 3dy/dx = 3x + 3 never differentiate a quotient like this directly — split into separate terms first!

Practice questions

Try each one yourself first, then click the question to reveal the worked answer. Aim to do all 5 in under 5 minutes.

Q1 EASY Differentiate y = 7x³ − 4x² + 5 Show answer ▼Hide answer ▲

7x³ → 21x² −4x² → −8x 5 → 0 dy/dx = 21x² − 8x

Q2 EASY Differentiate y = 6x⁵ + 2x − 9 Show answer ▼Hide answer ▲

6x⁵ → 30x⁴ 2x → 2 −9 → 0 dy/dx = 30x⁴ + 2

Q3 MEDIUM Differentiate y = 4/x³ Show answer ▼Hide answer ▲

rewrite first y = 4x⁻³ 4x⁻³ → −3·4x⁻⁴ = −12x⁻⁴ dy/dx = −12/x⁴

Q4 MEDIUM Differentiate y = 9√x − 2x² Show answer ▼Hide answer ▲

rewrite the root y = 9x^(1/2) − 2x² 9x^(1/2) → ½·9 x^(−1/2) = (9/2)x^(−1/2) −2x² → −4x dy/dx = 9/(2√x) − 4x

Q5 HARD Differentiate y = (4x² − 5x) / x Show answer ▼Hide answer ▲

split the fraction first y = 4x²/x − 5x/x = 4x − 5 4x → 4 −5 → 0 dy/dx = 4 never quotient-rule it — split first!

⚠ Common mistakes

Forgetting that d/dx[constant] = 0. The “−1” or “+5” at the end disappears, not stays.
Not rewriting fractions or roots as powers first — 6/x² must become 6x⁻² before you can apply the rule.
Sign errors with negative powers — x⁻² differentiates to −2x⁻³, not −2x⁻¹.
Forgetting to multiply BOTH the coefficient AND the power. 5x³ → 15x², not 3x².
Trying to differentiate a product or quotient directly — expand or split first when possible.

📖

Want the theory?

Read Differentiating Powers of x for the full breakdown, or Introduction to Derivatives for the conceptual basis.

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AA SL Differentiating Polynomials skills

The Method

Worked examples

Practice questions

⚠ Common mistakes

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