IB Maths AI HL
Further Differentiation
Paper 1 & 2
~7 min read
Differentiating Special Functions
The power rule handles polynomials, but exams also throw sin, cos, tan, ex and ln x at you. Each has its own standard derivative — most are in the formula booklet — and when the inside is a linear expression like ax + b, you simply multiply by the constant a. Get these results into muscle memory and a whole new class of questions opens up.
📘 What you need to know
- Trig derivatives: sin → cos, cos → −sin, tan → 1cos²x. All three are in the formula booklet.
- Exponential: y = ex differentiates to itself, ex.
- Logarithm: y = ln x gives 1x (for x > 0).
- Linear inside (ax + b): multiply the derivative by a.
- Radians only: trig calculus needs your GDC in radian mode.
- General inside f(x): multiply by f′(x) — that’s the chain rule, coming next.
Differentiating trig functions
The three basic trig derivatives are given to you in the formula booklet, so the skill is recognising them and handling the inside function. Always work in radians.
Standard trig derivatives
y = sin x → dydx = cos x · y = cos x → −sin x · y = tan x → 1cos²x
✓ all three given in the formula booklet
| Function | Linear inside (ax+b) | General inside f(x) |
|---|
| sin | a cos(ax+b) | f′(x) cos(f(x)) |
| cos | −a sin(ax+b) | −f′(x) sin(f(x)) |
| tan | acos²(ax+b) | f′(x)cos²(f(x)) |
🧠 “Sine to cosine, cosine flips sign”
sin differentiates to cos (no sign change); cos differentiates to −sin (a minus appears). The minus lives with the cosine. Whatever’s inside, multiply by its derivative.
⚠ Radians, every time
- These derivative results are only valid in radians. The moment you see trig differentiation, switch your GDC to radian mode or every value will be wrong.
Differentiating ex & ln x
Exponentials and logarithms have two derivatives worth knowing cold. The exponential is unique — it’s its own derivative.
Exponential & logarithm
y = ex → dydx = ex · y = ln x → 1x
✗ not in the booklet — learn these two
| Function | Linear inside (ax+b) | General inside f(x) |
|---|
| e(…) | a e(ax+b) | f′(x) ef(x) |
| ln(…) | aax+b | f′(x)f(x) |
🤔 Why is the derivative of ln kx still just 1x?
Using the linear rule, ln(kx) differentiates to kkx — and the k‘s cancel, leaving 1x. The constant inside a log simply doesn’t survive differentiation. (It makes sense via log laws too: ln kx = ln k + ln x, and ln k is a constant that differentiates to zero.)
⚠ Two classic slips
- The derivative of ln kx is 1x, NOT kx.
- The derivative of ekx is kekx, NOT kxekx−1 — don’t apply the power rule to an exponential.
Worked examples
WE 1Differentiate f(x) = sin x
The basic result, straight from the booklet.
f(x) = sin x
f′(x) = cos x
WE 2Differentiate f(x) = cos(5x + 1)
Linear inside: a = 5. Use −a sin(ax+b).
inside = 5x + 1, so a = 5
f′(x) = −5 sin(5x + 1)
WE 3A curve has equation y = tan(6x2 − π4). Find the gradient at x = √π2 (exact value).
General inside f(x) = 6x2 − π4, so f′(x) = 12x. Use f′(x)cos²(f(x)).
dy/dx = 12xcos²(6x² − π/4)
At x = √π/2: 6x² = 6·π4 = 3π2, so inside = 3π2 − π4 = 5π4
numerator 12x = 12·√π2 = 6√π; cos²(5π4) = (−1√2)² = 12
dy/dx = 6√π1/2 = 12√π
gradient = 12√π
WE 4A curve is y = e−3x+1 + 2 ln 5x. Find the gradient at x = 2 in the form a + bec.
Differentiate each term: linear exponential (×−3) and the special ln case.
dy/dx = −3e−3x+1 + 2·1x
(ln 5x → 1/x, the special b = 0 case)
At x = 2: −3e−3(2)+1 + 22 = −3e−5 + 1
gradient = 1 − 3e⁻⁵ (a = 1, b = −3, c = −5)
WE 5Spot the common slip: differentiate y = ln 7x and y = e4x
Apply the linear-inside rules carefully.
ln 7x → 77x = 1x (the 7’s cancel)
e4x → 4e4x (multiply by a = 4)
ln 7x → 1/x · e⁴ˣ → 4e⁴ˣ
NOT 7/x, and NOT 4x·e^(4x−1).
💡 Top tips
- GDC to radians the instant you see trig differentiation.
- The trig derivatives are in the booklet — but ex and ln x are not, so memorise those.
- Linear inside: just multiply by the constant a. General inside: multiply by f′(x).
- ln of any multiple of x gives 1x — the constant cancels.
- Exact-value questions often want surds or e to a power; keep π, √ and e symbolic rather than decimalising.
- Check on the GDC — it can evaluate a derivative at a point even if it won’t give exact form.
⚠ Common mistakes
- Working in degrees — the derivative formulas only hold in radians.
- Dropping the sign on cos — cos x differentiates to −sin x.
- ln kx → kx — wrong; the constant cancels to give 1x.
- Power-rule-ing an exponential — ekx is kekx, not kxekx−1.
- Forgetting the inside derivative — for sin/cos/tan/e/ln of a function, you must multiply by f′(x).
Next up — Chain Rule. You’ve already met the pattern: when the inside of sin, cos, e or ln is a function of x, you multiply by its derivative f′(x). That “multiply by the derivative of the inside” move is the chain rule — the next topic makes it explicit and shows how to handle any composite “function of a function”, not just these special ones.
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