IB Maths AA SL Topic 5 — Calculus Paper 1 & 2 🎯 Skill ~4 min practice

AA SL Product Rule skills

When you have two functions multiplied together — like x² · sin x or e^x · ln x — you can’t differentiate them separately. The product rule glues them with a clean two-piece formula. Set up u and v in a small table and the rest is mechanical.

The Method

dydx = u dvdx + v dudx “first × derivative of second + second × derivative of first” · ✓ in formula booklet

Split the product — call the first function u and the second v. Either order works.
Differentiate both pieces separately — find u‘ and v‘ in a small table.
Plug into uv‘ + vu‘, then simplify (factor common terms if asked).

Set up u and v in a table

Example: differentiate y = x² · sin x

	u (first)	v (second)
function	x²	sin x
derivative	2x	cos x

dy/dx = uv‘ + vu‘ = x² cos x + 2x sin x

Build the table first, then read off the formula. No algebra, no surprises.

Product rule vs Chain rule

✓ USE product rule when

Two functions are multiplied: f(x) · g(x)

e.g. x² sin x, e^x ln x, (2x+1)(x−3)

✗ USE chain rule instead when

One function is inside another: f(g(x))

e.g. (3x+1)⁵, sin(2x), e^x²

Worked examples

WE 1 EASY

Differentiate y = x³(2x − 5).

step 1 — set up u and v u = x³ v = 2x − 5 u’ = 3x² v’ = 2step 2 — apply uv’ + vu’ dy/dx = x³(2) + (2x − 5)(3x²)step 3 — simplify = 2x³ + 6x³ − 15x² = 8x³ − 15x²dy/dx = 8x³ − 15x² could also expand first to 2x⁴ − 5x³, then differentiate to get the same answer — pick whichever is faster!

WE 2 MEDIUM

Differentiate y = 4x · cos x.

step 1 — table u = 4x v = cos x u’ = 4 v’ = −sin xstep 2 — apply formula dy/dx = 4x(−sin x) + (cos x)(4)dy/dx = 4 cos x − 4x sin x d/dx(cos x) = −sin x — keep the minus sign carefully when expanding!

WE 3 HARD

Differentiate y = x² · e^3x. Factor the answer.

step 1 — set up u = x² → u’ = 2x v = e^3x → v’ = 3e^3x (chain rule!)step 2 — apply uv’ + vu’ dy/dx = x²(3e^3x) + e^3x(2x) = 3x²e^3x + 2xe^3xstep 3 — factor common terms common factor: x · e^3xdy/dx = x e^3x(3x + 2) when “factor the answer” is asked — pull out everything that’s common across both pieces!

Practice questions

Try each one yourself first, then click the question to reveal the worked answer. Build the u/v table before plugging in — saves errors every time.

Q1 EASY Differentiate y = x · ln x. Show answer ▼Hide answer ▲

u = x → u’ = 1 v = ln x → v’ = 1/x dy/dx = x(1/x) + (ln x)(1) dy/dx = 1 + ln x

Q2 EASY Differentiate y = (2x + 1)(x − 3). Show answer ▼Hide answer ▲

u = 2x + 1 → u’ = 2 v = x − 3 → v’ = 1 dy/dx = (2x + 1)(1) + (x − 3)(2) = 2x + 1 + 2x − 6 dy/dx = 4x − 5 expanding first then differentiating is faster here — both methods work!

Q3 MEDIUM Differentiate y = x² · sin x. Show answer ▼Hide answer ▲

u = x² → u’ = 2x v = sin x → v’ = cos x dy/dx = x² cos x + sin x · 2x dy/dx = x² cos x + 2x sin x

Q4 MEDIUM Differentiate y = e^x(x² + 1). Show answer ▼Hide answer ▲

u = e^x → u’ = e^x v = x² + 1 → v’ = 2x dy/dx = e^x(2x) + (x² + 1)(e^x) factor e^x dy/dx = e^x(x² + 2x + 1) = e^x(x + 1)² factoring reveals a perfect square hiding inside — bonus simplification!

Q5 HARD Differentiate y = (3x − 1)² · ln x. Factor the answer. Show answer ▼Hide answer ▲

u and v setup (chain rule on u) u = (3x − 1)² → u’ = 2(3x − 1)(3) = 6(3x − 1) v = ln x → v’ = 1/x apply uv’ + vu’ dy/dx = (3x − 1)²(1/x) + ln x · 6(3x − 1) factor (3x − 1) dy/dx = (3x − 1)[(3x − 1)/x + 6 ln x] always look for the common factor — it’s almost always there in IB questions!

⚠ Common mistakes

“Differentiating both” instead of using the rule. d/dx(uv) is NOT u‘v‘. The correct answer always has two terms added together.
Mixing up product and chain rule. Product is for u · v (multiplication). Chain is for u(v(x)) (one inside the other).
Sign errors with cos. d/dx(cos x) = −sin x. The minus stays through the formula.
Forgetting chain rule on u or v. If u = e^3x, then u‘ = 3e^3x (chain rule). Don’t drop the 3.
Not factoring when asked. “Factor the answer” needs you to pull out common terms — the marker wants the cleaner form.
Treating (2x + 1)(x − 3) as needing product rule. If you can expand first and skip product rule, do that — much faster on Paper 1.

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Want the theory?

Read the full Product Rule notes for the proof, the link to the quotient rule, and worked exam-style problems combining product rule with chain rule.

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