IB Maths AI HL
Further Differentiation
Paper 1 & 2
~7 min read
Chain Rule
In the last topic you kept multiplying by “the derivative of the inside”. The chain rule is the formal version of that move, and it works for any composite function — a “function of a function” like (3x+2)5 or sin(e2x), not just the special ones. Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
📘 What you need to know
- The rule: if y = g(u) and u = f(x), then dydx = dydu × dudx.
- It’s in the formula booklet in exactly that form.
- When to use it: composite functions — the variable doesn’t “appear alone”.
- The mantra: differentiate the outside (inside untouched), then × derivative of the inside.
- Substitution method: set u = inside, differentiate each part, multiply, swap u back.
- Aim to do it mentally — spotting it is faster than writing out the substitution every time.
Spotting a composite function
The chain rule is for a “function of a function”. The tell-tale sign is that x doesn’t appear on its own — something is done to x before the outer function is applied.
composite — use chain rule
sin(3x + 2)
x is tripled and has 2 added before sine is applied.
not composite
sin x
x “appears alone” — a plain standard derivative.
The chain rule
dydx = dydu × dudx or y = g(f(x)) → dydx = g′(f(x))f′(x)
✓ given in the formula booklet
🧠 “Outside first, then the inside’s derivative”
Say it as you go: “differentiate the outer function, ignore the inside, then multiply by the derivative of the inside.” That single sentence is the whole rule.
Using the chain rule
🧭 Recipe — the substitution method
- Identify the two functions: write y = g(u) and u = f(x) (the inside).
- Differentiate y w.r.t. u to get dydu.
- Differentiate u w.r.t. x to get dudx.
- Multiply using dydx = dydu × dudx, then substitute f(x) back in for u.
Five standard results follow straight from the chain rule — worth recognising on sight:
| If y = … | then dydx = … |
|---|
| (f(x))n | n f′(x) (f(x))n−1 |
| ef(x) | f′(x) ef(x) |
| ln(f(x)) | f′(x)f(x) |
| sin(f(x)) | f′(x) cos(f(x)) |
| cos(f(x)) | −f′(x) sin(f(x)) |
Trickier problems may need the chain rule applied more than once — for instance when the inside is itself a composite function. Peel one layer at a time.
Worked examples
WE 1Differentiate y = (x2 − 5x + 7)7
Power of a function. Set u = x2 − 5x + 7.
y = u⁷, u = x² − 5x + 7
dy/du = 7u⁶, du/dx = 2x − 5
dy/dx = 7u⁶(2x − 5), sub u back
dy/dx = 7(2x − 5)(x² − 5x + 7)⁶
WE 2Differentiate y = sin(e2x)
“sin of something.” Differentiate sin (ignore the inside), then × derivative of e2x.
outside: sin → cos(e²ˣ)
inside: e²ˣ → 2e²ˣ
dy/dx = cos(e²ˣ) × 2e²ˣ
dy/dx = 2e²ˣ cos(e²ˣ)
WE 3Differentiate y = (3x + 2)5
Use the standard result n f′(x)(f(x))n−1 with f(x) = 3x+2.
n = 5, f′(x) = 3
dy/dx = 5 · 3 · (3x + 2)⁴
dy/dx = 15(3x + 2)⁴
WE 4Differentiate y = ln(x2 + 1)
“ln of a function” → f′(x)f(x).
f(x) = x² + 1, f′(x) = 2x
dy/dx = 2xx² + 1
WE 5Differentiate y = cos3x (i.e. (cos x)3) — chain rule twice
This is a power of a function, where the function is itself cos x. Peel the outer power, then differentiate cos x.
y = (cos x)³, outer power: 3(cos x)²
inside cos x → −sin x
dy/dx = 3(cos x)² × (−sin x)
dy/dx = −3 cos²x sin x
💡 Top tips
- The rule’s in the booklet — but learn the five standard results so you can apply it instantly.
- Say the mantra: outside first (inside untouched), then × derivative of the inside.
- Aim to go mental — for simple composites, skip writing u and just spot the layers.
- Layer by layer — if the inside is also composite, apply the chain rule again.
- Radians for any trig, as always.
- Don’t confuse it with the product rule — sin(cos x) is composite (chain); sin x cos x is a product.
⚠ Common mistakes
- Forgetting the inside derivative — (3x+2)5 → 5(3x+2)4 is wrong; you must also × 3.
- Differentiating the inside too — leave the inside untouched in the outer part; the inside’s derivative is a separate multiplying factor.
- Confusing composite with product — “function of a function” needs the chain rule, not the product rule.
- Stopping after one layer — nested composites need the chain rule more than once.
- Dropping a sign — cos differentiates to −sin, so watch the minus when cos is involved.
Next up — Product Rule. The chain rule handles a function inside another. The product rule handles a different situation: two functions multiplied together, like ex sin x. Watch for the easy mix-up — sin(cos x) is composite (chain rule), but sin x cos x is a product (product rule) — and note that trickier questions will need the chain rule inside the product rule.
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