IB Maths AA Calculus: Limits, Derivatives & Differential Equations

Q: Which differentiation topics are HL-only?

HL adds derivatives of $a^x$, $\log_a x$, and inverse trig functions, plus implicit differentiation and related rates problems, on top of the SL rule toolkit.

The single biggest HL topic — here's what actually shows up in Paper 1, Paper 2 and long-response questions

A curve with a tangent line at a point, alongside a shrinking secant line approaching it to represent a limit

Subject

Maths AA

Curriculum

IB Diploma Programme

Grade

Topic

Calculus

Reading

8 min

Difficulty

Advanced

Quick facts

Difficulty

★★★★☆

Exam weight

~26% of HL teaching hours

Prerequisites

Functions, trig identities, algebra

You'll learn

Limits, differentiation, optimisation, differential equations

Revision time

3-4 hours for a solid first pass

Calculus is the backbone of IB Maths AA HL — it makes up around a quarter of all HL teaching hours and almost always anchors a Section B long-response question. Everything traces back to one idea: the limit. Continuity, differentiation, and integration are all defined using limits, which is why examiners love testing whether you actually understand the definition, not just the shortcut rules. This teaser walks through the five ideas that come up again and again: the three-condition continuity test, the derivative as a limit, the HL-only differentiation toolkit (implicit differentiation, inverse trig, related rates), stationary points and optimisation, and the leap into HL differential equations. Each one carries its own classic examiner trap. Master these five, and you'll be ready for the calculus questions that decide grades — the full revision notes go much deeper into every worked example and technique.

What you’ll be able to do

✓State and apply the three conditions for continuity at a point

✓Evaluate limits using factoring, conjugates, standard trig/exponential limits, and the squeeze theorem

✓Differentiate from first principles using the limit definition

✓Apply the chain, product, and quotient rules without mixing them up

✓Differentiate inverse trig functions and use implicit differentiation (HL)

✓Solve related rates problems by chaining two rates together

✓Locate and classify stationary points and points of inflection

✓Set up and solve optimisation problems by eliminating a variable first

Limits and the Three-Condition Continuity Test

A limit describes what $f(x)$ approaches as $x$ gets arbitrarily close to a value, regardless of what happens exactly at that point. IB doesn't ask for formal proofs — it wants algebraic evaluation (factor and cancel, rationalise with a conjugate, or use a standard trig/exponential limit) and reasoning about continuity. A function is continuous at $x=a$ only if all three conditions hold: $f(a)$ exists, the two-sided limit exists, and the limit equals $f(a)$ .

A piecewise function graph with a jump at x=a, showing the left-hand limit and right-hand limit as separate arrows

0/0 situation	Technique
Polynomial fraction	Factor and cancel the common root
Surd expression	Multiply by the conjugate
sin, tan, arctan, or e^x-1 form	Engineer into a standard limit shape
Bounded oscillating factor × something → 0	Squeeze theorem

Exam tip

For 'show that' limit questions, you must factor/rationalise/rewrite algebraically before substituting — typing the unsimplified fraction into a GDC and stating the answer scores 0/2, even if the number is right.

Common mistake

Treating 'the two branch limits agree' as the whole continuity check. Always write out all three conditions explicitly, even if the algebra collapses two of them into one equation.

Mini summary

Continuity needs all three conditions to hold; standard limits and the squeeze theorem solve almost every trig/exponential 0/0 case.

The Derivative as a Limit, Then the Rule Toolkit

The derivative is officially defined as $f'(a) = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}$ , and you're expected to use this definition directly whenever a question demands it — it's the only legal method at points where shortcut rules don't apply. Everywhere else, you reach for the chain rule for composite functions, and the product or quotient rule for two functions multiplied or divided together. Mixing up the product and quotient rule under time pressure is the single most common HL differentiation error.

A diagram showing the difference quotient f(a+h)-f(a) over h as h shrinks toward zero, converging to the tangent slope

Exam tip

Say the rule out loud as you write it ("derivative of first times second, plus first times derivative of second") to stop product/quotient signs blurring together under exam pressure.

Common mistake

Forgetting the chain rule inside a composite trig function, or swapping the sign pattern between the product and quotient rules.

Mini summary

Use the first-principles definition when told to; otherwise apply chain, product, and quotient rules carefully and deliberately.

HL-Only Extras: Inverse Trig, Implicit Differentiation, Related Rates

HL adds derivatives of $a^x$ , $\log_a x$ , and all three inverse trig functions to the SL toolkit. Implicit differentiation lets you differentiate both sides of an equation in $x$ and $y$ with respect to $x$ , treating $y$ as a function of $x$ throughout. Related rates problems chain two rates of change together, such as $\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}$ , so you can find how fast one quantity changes from how fast another one does.

A cone-shaped tank filling with water, with labelled rates dV/dt and dh/dt connected by a chain-rule arrow

Exam tip

In related rates problems, identify every rate given and every rate asked for first, then decide which chain-rule link connects them — writing the template before substituting numbers avoids sign and unit errors.

Mini summary

HL differentiation adds inverse trig derivatives, implicit differentiation, and related-rates chaining on top of the SL rule set.

Stationary Points, Inflection, and Optimisation

A stationary point occurs where $f'(x)=0$ , giving a horizontal tangent that could be a local max, local min, or horizontal inflection point. A point of inflection is where concavity changes — where $f''(x)$ changes sign either side of it, so the tangent crosses the curve. Optimisation problems (like minimising the surface area of a can with fixed volume) require you to eliminate one variable using the given constraint before differentiating.

A curve showing a local maximum, local minimum, and inflection point, each labelled with the sign of f' and f''

Exam tip

Substitute the constraint to eliminate a variable first — differentiating a two-variable expression while silently holding one variable 'constant' is the most common optimisation error at HL.

Common mistake

Differentiating an expression like surface area while forgetting to substitute the volume constraint first, which leaves an extra variable hiding in the derivative.

Mini summary

Stationary points and inflection points come from $f'$ and $f''$ respectively; always eliminate a variable using the constraint before optimising.

Integration and HL Differential Equations

Integration is the reverse of differentiation, and once you can integrate, you can find areas and volumes generated by curves. HL pushes this further into differential equations — equations involving $\frac{dy}{dx}$ itself — which you solve by separating variables, using substitution, or applying an integrating factor. This strand ties derivatives and integrals together to model how quantities change over time, and it's a favourite source of HL-only long-response questions.

A region under a curve shaded to represent an integral, next to a small differential equation dy/dx = ky being solved by separating variables

Mini summary

Integration reverses differentiation to find areas and volumes; HL differential equations combine both skills to model change.

Quick formula sheet

\lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1

The small-angle sine limit, the backbone of every trig limit question. — Match the angle inside sin to the denominator, then this limit is automatically 1.

\lim_{x \to 0} \frac{e^{x}-1}{x} = 1

Standard exponential limit used whenever an exponential parameter tends to 0.

\lim_{x \to 0} \frac{\arctan x}{x} = 1

HL arctan version of the same standard-limit trick.

f'(a) = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}

The derivative at a point, defined as a limit of a difference quotient.

\frac{d}{dx}(x^n) = nx^{n-1}

Power rule for differentiation.

(uv)' = u'v+uv'

Product rule for differentiating two multiplied functions.

\left(\frac{u}{v}\right)' = \frac{u'v-uv'}{v^2}

Quotient rule for differentiating two divided functions.

\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}

Chain rule for differentiating composite functions.

\frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}

HL-only derivative of arctan, one of the three inverse trig derivatives.

\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}

The related rates chain-rule template used to connect two changing quantities (HL).

Practice questions

Easy

Evaluate $\lim_{x\to 3} \frac{x^2-9}{x-3}$ by factoring first.
State the three conditions required for a function to be continuous at $x=a$ .
Differentiate $y = x^3\sin x$ using the product rule.

Medium

Use the squeeze theorem to find $\lim_{x\to 0} x^2\cos\left(\frac{1}{x}\right)$ , stating the bounding inequality explicitly.
Find $\frac{dy}{dx}$ for $x^2+y^2=25$ using implicit differentiation.
A stone dropped into a pond creates a circular ripple whose radius grows at 3 cm/s. Find the rate of change of the area when the radius is 10 cm.

Challenge

For $f(x)=\frac{\sin(ax)}{x}+b$ on $x<0$ and $f(x)=\frac{e^{cx}-1}{x}$ on $x>0$ with $f(0)=k$ , find the relationship between $a$ , $b$ , $c$ , and $k$ for continuity at 0.
A closed cylindrical can must hold $500\text{ cm}^3$ . Find the radius that minimises total surface area and justify that it is a minimum.
Using the definition of the derivative, find $f'(0)$ for $f(x)=\frac{\sin(2x)}{x}$ (with $f(0)$ defined for continuity), without using L'Hôpital's rule.

Frequently asked questions

How much of IB Maths AA HL is calculus?+

Calculus makes up roughly 26% of HL teaching hours, making it the single largest HL topic. It's examined across both Paper 1 and Paper 2 and almost always anchors at least one Section B long-response question.

What are the three conditions for continuity in IB Maths AA?+

A function is continuous at $x=a$ if $f(a)$ exists, the two-sided limit at $x=a$ exists, and that limit equals $f(a)$ . All three must hold — checking only that the branch limits agree isn't enough.

Is differentiability the same as continuity?+

No. Differentiable functions are always continuous, but continuous functions aren't always differentiable — corner points are the classic HL example where a function is continuous but not differentiable.

Which differentiation topics are HL-only?+

HL adds derivatives of $a^x$ , $\log_a x$ , and inverse trig functions, plus implicit differentiation and related rates problems, on top of the SL rule toolkit.

What's the biggest mistake students make in optimisation problems?+

Forgetting to substitute the given constraint to eliminate a variable before differentiating. Differentiating with a hidden extra variable is the most common HL optimisation error.

Are differential equations examined at SL?+

No, differential equations are an HL-only strand. You solve them by separating variables, substitution, or using an integrating factor, combining derivative and integral skills together.

Get the full IB Maths AA HL Calculus revision notes

Complete worked examples for every limit, derivative, and differential equation technique in this teaser Full breakdowns of examiner traps with step-by-step fixes A complete formula sheet plus original mock papers and exam-style questions to test yourself

Get the Calculus notes on RevisionPrep →

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