IB Maths AA SL Topic 5 — Calculus Paper 1 & 2 ~8 min read

Chain Rule

Use the chain rule whenever you have a “function inside a function” — like (3x + 1)⁵ or sin(x²) or e^cosx. The trick: differentiate the outside, then multiply by the derivative of the inside. That’s it.

📘 What you need to know

The rule: dydx = dydu × dudx (in formula booklet).
Use it when the variable doesn’t “appear alone” — e.g. (2x − 1)⁴, ln(x² + 5), e^3x.
Mental shortcut: outside derivative × inside derivative. No u substitution needed once you spot the pattern.
Sometimes you need to apply the chain rule twice in one expression.

When to use it

Composite functions need the chain rule. The variable x is “wrapped inside” something else.

⚠ Not composite

sin x

x appears alone — just use the standard rule

✓ Composite — use chain rule

sin(3x + 2)

x is wrapped inside (3x + 2) first

The formula

Chain rule

If y = g(u) and u = f(x) then dydx = dydu × dudx

✓ in formula booklet

The 3-step method

How to apply the chain rule

Set u = inside function, so y = outside function of u.
Differentiate both: find dy/du and du/dx.
Multiply: dy/dx = dy/du × du/dx, then sub u back in.

The 5 standard results (chain rule in action)

These are what the chain rule gives for the most common composite functions. Memorise the patterns — you’ll skip the u-substitution most of the time.

If y =	Then dy/dx =
(f(x))ⁿ	n · f′(x) · (f(x))ⁿ⁻¹
e^f(x)	f′(x) · e^f(x)
ln(f(x))	f′(x) / f(x)
sin(f(x))	f′(x) · cos(f(x))
cos(f(x))	−f′(x) · sin(f(x))

The mental shortcut

What to say in your head

differentiate outside × differentiate inside

“Keep the inside the same, derivative of outside × derivative of inside”

🧠

“Outside × inside”

For sin(x²): outside is sin → cos. Inside is x² → 2x. Answer: cos(x²) × 2x = 2x cos(x²). Done in one line.

📍

Sometimes chain rule applies twice

For sin(e^2x): outside is sin → cos. Inside is e^2x — but THAT also needs chain rule → 2e^2x. Final: 2e^2x cos(e^2x).

Worked examples

WE 1

Power of a function

Find the derivative of y = (x² − 5x + 7)⁷.

step 1 — set u u = x² − 5x + 7, y = u⁷step 2 — differentiate both dy/du = 7u⁶, du/dx = 2x − 5step 3 — multiply & sub back dy/dx = 7u⁶ × (2x − 5) = 7(2x − 5)(x² − 5x + 7)⁶ dy/dx = 7(2x − 5)(x² − 5x + 7)⁶ classic pattern: n · (inside)′ · (inside)ⁿ⁻¹

WE 2

Trig with a power inside

Differentiate y = cos(x³).

Outside: cos → −sin Inside: x³ → 3x² Multiply: dy/dx = −sin(x³) × 3x² = −3x² sin(x³) dy/dx = −3x² sin(x³) keep the inside intact (x³) inside the sin — never simplify it!

WE 3

e to a function

Differentiate y = e^{x² + 1}.

Outside: e^{( )} → e^{( )} (stays the same!) Inside: x² + 1 → 2x Multiply: dy/dx = 2x · e^{x² + 1} dy/dx = 2x e^{x² + 1} e^(anything) stays as e^(anything) — that’s the easy bit!

WE 4

ln of a function

Differentiate y = ln(3x² − 4x).

Outside: ln( ) → 1/( ) Inside: 3x² − 4x → 6x − 4 Pattern: f′(x)/f(x): dy/dx = (6x − 4)/(3x² − 4x) “derivative of inside, over the inside” — that’s the ln pattern!

WE 5

Chain rule applied twice

Find the derivative of y = sin(e^2x).

Three layers: sin( e( 2x ) ). Peel from outside in.Outermost: sin → cos Keep the inside e^2x intact: cos(e^2x) Multiply by derivative of e^2x: e^2x → 2 e^2x (chain rule again!) dy/dx = cos(e^2x) × 2 e^2x dy/dx = 2 e^2x cos(e^2x) two applications: once for sin(__), once for e^2x.

💡 Top tips

Spot the inside function first. What’s wrapped up that doesn’t appear alone? That’s u.
“Outside × inside” — derivative of outer × derivative of inner. Keep saying it.
Keep the inside untouched in the outside derivative. cos(x³) → −sin(x³), NOT −sin(3x²).
Use the 5 standard results as a shortcut — no need to write u = … once you see the pattern.
Check for nested chains. If the inside itself has an inside, apply chain rule twice.
Always work in radians for trig.

⚠ Common mistakes

Forgetting to multiply by the derivative of the inside. The most common chain rule error.
Differentiating the inside when you shouldn’t. cos(x²) becomes −sin(x²), not −sin(2x).
Confusing composite with product. sin(cos x) is composite (chain rule). sin x · cos x is a product (different rule, next note).
Stopping after one application when chain rule was needed twice (like WE 5).
Sign errors on cos. cos goes to negative sin. Always.

Chain rule is for “function of a function”. The next note covers the product rule — for when two functions are multiplied together. Don’t mix them up.

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