Differentiation is the single most heavily examined topic in IB Mathematics Analysis & Approaches HL, appearing on every Paper 1, Paper 2, and Paper 3. The HL syllabus extends the SL content (basic rules, optimisation, kinematics) with four high-leverage tools — L'Hôpital's rule, implicit differentiation, related rates, and the extended derivative table — and these account for the majority of the marks lost by HL students.
This cheatsheet condenses every formula, trick, and trap from SL 5.1–5.8 and AHL 5.12–5.15 into one page you can revise from. If you want the printable PDF version, the full set of notes, the worked tutorials, and the marked-up solutions, scroll to the bottom for the download links and the gated full library.
§1 — Limits & First Principles SL 5.1, AHL 5.12
Limit definition
Key limits to memorise
§2 — Basic Differentiation Rules SL 5.3, 5.6
Core rules
Standard derivatives (SL)
| $f(x)$ | $f'(x)$ |
|---|---|
| $e^x$ | $e^x$ |
| $\ln x$ | $\dfrac{1}{x}$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\sin(ax+b)$ | $a\cos(ax+b)$ |
| $e^{ax+b}$ | $a e^{ax+b}$ |
| $\ln(ax)$ | $\dfrac{1}{x}$ (the $a$ cancels!) |
§3 — Extended HL Derivatives AHL 5.15
Reciprocal trig (NOT in the formula booklet — memorise!)
| $f(x)$ | $f'(x)$ |
|---|---|
| $\tan x$ | $\sec^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
| $\cot x$ | $-\csc^2 x$ |
Memory aid: all the co-functions ($\cos$, $\csc$, $\cot$) carry a minus sign in their derivative.
Exponential, log & inverse trig (in the formula booklet)
| $f(x)$ | $f'(x)$ |
|---|---|
| $a^x$ | $a^x \ln a$ |
| $\log_a x$ | $\dfrac{1}{x \ln a}$ |
| $\arcsin x$ | $\dfrac{1}{\sqrt{1 - x^2}}$ |
| $\arccos x$ | $-\dfrac{1}{\sqrt{1 - x^2}}$ |
| $\arctan x$ | $\dfrac{1}{1 + x^2}$ |
§4 — Higher Derivatives AHL 5.12
$f''(x) = \dfrac{d^2 y}{dx^2}$ — second derivative. $f^{(n)}(x) = \dfrac{d^n y}{dx^n}$ — $n$-th derivative. Differentiate $f'(x)$ to get $f''(x)$; repeat for higher orders.
Concavity from $f''$: $f''(x) > 0$ → concave up (smile); $f''(x) < 0$ → concave down (frown); $f''(x) = 0$ AND sign change → point of inflexion.
§5 — Implicit Differentiation AHL 5.14
Key rule: $\dfrac{d}{dx}[f(y)] = f'(y) \cdot \dfrac{dy}{dx}$.
4-step method
- Differentiate every term with respect to $x$.
- Apply the product rule to any $xy$-terms.
- Collect every $\dfrac{dy}{dx}$ on one side.
- Factorise and divide.
Essential examples
§6 — L'Hôpital's Rule AHL 5.13
Only for $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ indeterminate forms:
$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$
Differentiate numerator and denominator separately. Repeat if still indeterminate.
IB mark scheme requirements
- State the indeterminate form (R1).
- Write $\lim$ notation at every step.
- Show derivatives of numerator AND denominator.
- Final evaluated value (A1).
Omitting $\lim$ notation loses the final A1.
§7 — Stationary Points & Optimisation SL 5.8
Classification
| $f'(a) = 0$ | $f''(a)$ | Type |
|---|---|---|
| $> 0$ | Local minimum | |
| $< 0$ | Local maximum | |
| $= 0$ | Use sign of $f'$ test |
Inflexion: $f'' = 0$ and sign changes. Increasing: $f' > 0$. Decreasing: $f' < 0$.
Optimisation 7-step method
- Define your variables.
- Write the objective function.
- Apply the constraint to eliminate one variable.
- Differentiate; set $= 0$.
- Classify (max or min).
- Check the domain endpoints.
- Answer what was actually asked.
§8 — Related Rates of Change AHL 5.14
Chain rule for rates: $\dfrac{dy}{dt} = \dfrac{dy}{dx} \cdot \dfrac{dx}{dt}$.
Common formulae
- Sphere: $V = \tfrac{4}{3}\pi r^3 \Rightarrow \dfrac{dV}{dt} = 4\pi r^2 \dfrac{dr}{dt}$
- Circle: $A = \pi r^2 \Rightarrow \dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt}$
- Equilateral triangle: $A = \tfrac{\sqrt{3}}{4} x^2 \Rightarrow \dfrac{dA}{dt} = \tfrac{\sqrt{3}}{2} x \dfrac{dx}{dt}$
5-step method
- Draw a diagram, label variables.
- Write the geometric formula.
- Differentiate both sides w.r.t. $t$.
- Find any unknown variable values first.
- Substitute rates and solve. Always state units.
§9 — Kinematics SL 5.8
$v(t) = s'(t)$ and $a(t) = v'(t) = s''(t)$.
| $v > 0$ | Moving in positive direction |
| $v < 0$ | Moving in negative direction |
| $v = 0$ | Stationary (check sign change) |
| $a > 0, v > 0$ or $a < 0, v < 0$ | Speeding up |
| $a > 0, v < 0$ or $a < 0, v > 0$ | Slowing down |
Displacement vs total distance
Displacement: $s(b) - s(a)$ (signed). Total distance: split at every $t$ where $v = 0$, then add $|\Delta s|$ for each segment. On the GDC (P2): nInt(abs(v(t)),t,a,b). Speed $= |v(t)|$.
§10 — Tangents & Normals SL 5.4
Tangent at $(a, f(a))$: gradient $= f'(a)$ → $y - f(a) = f'(a)(x - a)$.
Normal at $(a, f(a))$: gradient $= -\dfrac{1}{f'(a)}$ → $y - f(a) = -\dfrac{1}{f'(a)}(x - a)$.
Tangent to a curve — discriminant method: if line $y = mx + c$ is tangent to $y = f(x)$, set them equal and demand $\Delta = 0$ (exactly one solution).
§11 — GDC Skills (TI-Nspire CX II) Paper 2
| Task | Syntax |
|---|---|
| $f'(a)$ | d(f(x),x)|x=a |
| $f''(x)$ | d(f(x),x,2) |
| Graph of $f'$ | f2(x)=d(f1(x),x) |
| Solve $f'(x) = k$ | solve(d(f(x),x)=k,x) |
| Total distance | nInt(abs(v(t)),t,a,b) |
| Max / min | Menu → Analyze Graph |
Common syntax traps: sinx → sin(x); e^x → e^(x); 3x → 3*x; sin^2(x) → (sin(x))^2; arctan(x) → atan(x).
§12 — Exam Attack Plan All sections
When you see this in the question — reach for that:
| Question trigger | Reach for |
|---|---|
| Polynomial, power, or rational | Power rule; rewrite roots/fractions as powers first |
| $f(g(x))$ — composite function | Chain rule; identify outer and inner |
| Two functions multiplied | Product rule: $u'v + uv'$ |
| $F(x, y) = c$, mixed equation | Implicit differentiation |
| $\frac{0}{0}$ or $\frac{\infty}{\infty}$ limit | State form (R1), then L'Hôpital |
| "Maximise / minimise" | Write objective, eliminate variable, differentiate, classify |
| "Rate of change / increasing at" | Related rates: chain rule for $t$ |
| "Total distance" | Split at $v = 0$; sum $|\Delta s|$ per segment |
| "Show that $\dfrac{dy}{dx} = \ldots$" | AG question — every step must be shown explicitly |
| First principles | Expand $f(x+h)$, cancel $h$, take $\lim_{h\to 0}$ |
Worked Example — IB-Style Implicit Differentiation
Question (HL Paper 1 style — 6 marks)
The curve $C$ is defined by $x^3 + y^3 = 6xy$ (the Folium of Descartes). Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$, and hence find the equation of the tangent to $C$ at the point $(3, 3)$.
Solution
- Differentiate both sides with respect to $x$ — apply the product rule to $6xy$:
$3x^2 + 3y^2 \dfrac{dy}{dx} = 6y + 6x\dfrac{dy}{dx}$ (M1)(A1) - Collect $\dfrac{dy}{dx}$ on one side: $3y^2 \dfrac{dy}{dx} - 6x\dfrac{dy}{dx} = 6y - 3x^2$ (M1)
- Factorise and divide: $\dfrac{dy}{dx} = \dfrac{6y - 3x^2}{3y^2 - 6x} = \dfrac{2y - x^2}{y^2 - 2x}$ (A1)
- Substitute $(3, 3)$: gradient $= \dfrac{2(3) - 9}{9 - 6} = \dfrac{-3}{3} = -1$ (A1)
- Tangent line: $y - 3 = -1(x - 3) \Rightarrow y = -x + 6$ (A1)
Examiner's note: Forgetting to apply the product rule on $6xy$ (writing just $6y$) is the single most common error here. The product rule gives $6y + 6x\dfrac{dy}{dx}$ — both terms are required.
Common Student Questions
What is the difference between $f''(x) = 0$ and a point of inflexion?
When can I use L'Hôpital's rule in IB Math AA HL?
Why do I keep losing marks on related rates questions?
How do I find total distance vs displacement in kinematics?
nInt(abs(v(t)),t,a,b). Confusing the two is the single most dropped mark in HL kinematics — Photon Academy students drill this in the first month.Do I need to memorise the reciprocal trig derivatives ($\sec$, $\csc$, $\cot$)?
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