Which differentiation rules must IB Maths AA students know?

Both SL and HL students need fluency with the power, chain, product, and quotient rules, together with derivatives of standard exponential, logarithmic, and trigonometric functions in their syllabus. You should recognize these rules from the structure of a function rather than wait for the question to name them. Composite functions require the chain rule, while products and quotients of variable expressions require their corresponding rules. Some questions combine several rules, such as differentiating $x^2e^{3x}$. HL students also study additional derivative forms and techniques specified in the HL syllabus. Use the official formula booklet during practice so that you know what is supplied and where to find it.

How much working should I show in an IB differentiation question?

Show enough working for an examiner to follow the mathematical method from the original function to the answer. For a routine derivative, one clear intermediate line may be sufficient, but a tangent, optimization, or stationary-point problem requires several connected stages. Write the derivative explicitly before substituting a value or solving an equation. Calculator output should not replace the setup, particularly when the question assesses reasoning or requires justification. If the command term is “show that,” your derivation must lead logically to the given result without assuming it. Clear working can also preserve method marks when a later arithmetic error occurs.

Are differentiation questions different on Paper 1 and Paper 2?

The underlying calculus is the same, but the permitted tools change how you approach the question. Paper 1 does not allow technology, so expressions are generally designed to be handled through exact algebra, known derivatives, and manageable equations. Paper 2 allows technology, which can support graphing, equation solving, numerical derivatives, and checking results. You must still show the mathematical setup and interpret calculator output appropriately. Practice some differentiation questions entirely by hand and others using the calculator functions permitted on Paper 2. HL Paper 3 may embed differentiation within a longer unfamiliar problem requiring several connected decisions.

How do I know whether to use the first or second derivative?

Use the first derivative when you need gradients, stationary points, increasing or decreasing intervals, or instantaneous rates of change. Solve $f'(x)=0$ to locate possible stationary points, then find their coordinates using the original function. The second derivative describes how the gradient changes and is commonly used for concavity or stationary-point classification. If $f''(a)>0$, a stationary point at $x=a$ is a local minimum, while $f''(a)<0$ indicates a local maximum. When $f''(a)=0$, the test is inconclusive rather than proof of an inflexion. In that case, examine sign changes in $f'$ or $f''$ around the point.

What is the best way to revise IB Maths AA differentiation questions?

Begin with a short diagnostic set covering direct differentiation, tangents, stationary points, optimization, and graph interpretation. Attempt each question before viewing its answer so that you expose the exact decision causing difficulty. Then watch a per-question worked video solution and compare its first few steps with your own method. Reproduce the solution without looking, followed by a different question using the same structure. Record mistakes under categories such as chain rule, algebra, classification, or interpretation, and revisit them after several days. Once individual methods are secure, complete mixed and timed questions so that you must choose the method independently.

What are the most important HL-only differentiation skills?

HL students extend the common SL foundation through topics that can include higher derivatives, continuity and differentiability, implicit differentiation, related rates, and further derivative forms. These skills are often combined with algebra, functions, trigonometry, or modelling rather than assessed in isolation. In implicit differentiation, every differentiated function of $y$ must include the factor $dy/dx$. Related-rates questions require a relationship between variables before differentiating with respect to time. Higher-level questions may also require interpretation, proof, or multiple linked steps rather than a single derivative. Check your teacher's current course plan and the official subject guide to distinguish required HL content from useful enrichment.

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IB Maths AA Differentiation Questions: Exam Guide

Q: What is the best way to revise IB Maths AA differentiation questions?

Begin with a short diagnostic set covering direct differentiation, tangents, stationary points, optimization, and graph interpretation. Attempt each question before viewing its answer so that you expose the exact decision causing difficulty. Then watch a per-question worked video solution and compare its first few steps with your own method. Reproduce the solution without looking, followed by a different question using the same structure. Record mistakes under categories such as chain rule, algebra, classification, or interpretation, and revisit them after several days. Once individual methods are secure, complete mixed and timed questions so that you must choose the method independently.

Q: What are the most important HL-only differentiation skills?

HL students extend the common SL foundation through topics that can include higher derivatives, continuity and differentiability, implicit differentiation, related rates, and further derivative forms. These skills are often combined with algebra, functions, trigonometry, or modelling rather than assessed in isolation. In implicit differentiation, every differentiated function of $y$ must include the factor $dy/dx$. Related-rates questions require a relationship between variables before differentiating with respect to time. Higher-level questions may also require interpretation, proof, or multiple linked steps rather than a single derivative. Check your teacher's current course plan and the official subject guide to distinguish required HL content from useful enrichment.

• 5 Aug 2026•14

IB Maths AA Differentiation Questions: Exam Guide

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IB Maths AA Differentiation Questions: Exam Guide | RevisionDojo

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IB Maths AA Differentiation Questions: Exam Guide

• 5 Aug 2026•14

Differentiation questions in IB Mathematics: Analysis and Approaches are usually built from a small number of recognizable structures. To answer them reliably, identify the function's structure, choose the appropriate rule, show the derivative clearly, and then interpret it in the context of gradients, stationary points, motion, or optimization.

The most efficient way to improve is not to reread differentiation notes repeatedly. Attempt an exam-style question first, watch a complete worked video solution, identify the first step where your method diverged, and then solve a similar question without support. This guide explains the recurring question types, the methods they require, and the traps that cost marks.

How differentiation is examined in IB Maths AA

The official IB Mathematics: Analysis and Approaches guide places differentiation within Topic 5: Calculus. The derivative is interpreted both geometrically, as the gradient of a curve, and physically, as an instantaneous rate of change.

Differentiation can appear in short-response or extended-response questions. Under the current assessment structure, Paper 1 does not permit technology, while Paper 2 permits technology; HL students also take a technology-allowed Paper 3 containing extended problem-solving questions. The official IB Maths AA subject brief confirms these paper formats.

At both SL and HL, questions may test differentiation directly or embed it within functions, graph analysis, kinematics, or modelling. HL students also encounter extensions such as higher derivatives, implicit differentiation, related rates, continuity and differentiability, and further derivative forms.

Question type	What you normally need to do	Frequent trap
Differentiate a function	Apply the correct rule and simplify	Missing an inner derivative
Tangent or normal	Find the point and gradient, then form a line equation	Using the tangent gradient for the normal
Stationary points	Solve $f'(x)=0$ and classify each point	Giving only the x-coordinate
Increasing or decreasing	Determine the sign of $f'(x)$ on intervals	Treating $f'(x)=0$ as the complete answer
Optimization	Build an objective function, differentiate, and justify the optimum	Ignoring constraints or endpoints
Kinematics	Connect displacement, velocity, and acceleration	Confusing position with distance travelled
Graph interpretation	Relate $f$ , $f'$ , and $f''$	Assuming $f^{''} (x) =$ guarantees an inflexion
Implicit differentiation	Differentiate both sides with respect to $x$	Omitting $dy/dx$ when differentiating a term in $y$

A reliable method for IB Maths AA differentiation questions

Use the following sequence before doing substantial algebra.

Read the final instruction first. Decide whether the question wants a derivative, coordinate, equation, interval, maximum value, or interpretation.
Identify the function's structure. Look for a composite, product, quotient, implicit relation, or expression that should be simplified first.
Choose the differentiation rule. Write enough intermediate working to make the method visible.
Use the derivative for the required purpose. For example, substitute a coordinate, solve $f'(x)=0$ , or analyze its sign.
Check restrictions and context. Reject values outside the domain and include units when a rate is involved.
State the requested conclusion. Do not stop at a derivative if the question asks for a maximum area or equation of a normal.

This separation between differentiating and using the derivative is important. Many students perform the calculus correctly but lose the final marks because they have not answered the actual question.

Recognizing the correct differentiation rule

The official formula booklet is available during examinations, but familiarity with its layout is essential. A formula being supplied does not remove the need to recognize when it applies.

Structure	Rule	Recognition clue
$x^n$	$d(x^n)/dx=nx^{n-1}$

For example, consider

$y=(x^2+1)e^x$ .

This is a product, so

$y'=2xe^x+(x^2+1)e^x$ .

Factorizing gives

$y'=e^x(x^2+2x+1)=e^x(x+1)^2$ .

The factorized form is particularly useful if the next part asks for stationary points or the sign of the derivative.

Worked differentiation question types

Tangents and normals

Suppose $f(x)=x^3-2x$ and the tangent is required at $x=1$ . First find the point:

$f(1)=1-2=-1$ .

Differentiate and evaluate the gradient:

$f'(x)=3x^2-2$ , so $f'(1)=1$ .

Using point-gradient form,

$y+1=1(x-1)$ ,

so the tangent is $y=x-2$ . A normal is perpendicular to the tangent, so its gradient is the negative reciprocal, $-1$ , giving $y+1=-(x-1)$ or .

A common error is to calculate the derivative correctly but substitute the x-coordinate into $f'$ without first finding the corresponding y-coordinate on the original curve.

Stationary points and classification

A stationary point occurs where $f'(x)=0$ . After solving this equation, substitute each x-value into the original function to obtain complete coordinates.

Classification can be completed using one of two standard approaches:

If $f''(a)>0$ , the point is a local minimum.
If $f''(a)<0$ , the point is a local maximum.

If $f''(a)=0$ , the second derivative test is inconclusive. It does not by itself prove that the point is an inflexion, so examine concavity or the sign of $f''$ on either side.

Optimization

Suppose a rectangle has perimeter 20 units. If one side is $x$ , the other is $10-x$ , so its area is

$A(x)=x(10-x)=10x-x^2$ , where $0<x<10$ .

Differentiate and solve:

$A'(x)=10-2x=0$ , giving $x=5$ .

Because $A''(x)=-2<0$ , this stationary value is a maximum. The maximum area is therefore $A(5)=25$ square units.

A complete optimization answer must include the objective function, its valid domain, the stationary-value calculation, justification that the result is a maximum or minimum, and the requested contextual value. RevisionDojo's guide to optimization with derivatives provides further structured examples.

Kinematics

If displacement is $s(t)$ , then velocity is $v(t)=s'(t)$ and acceleration is $a (t) = v^{'} (t) = s^{'}$ . For

$s(t)=t^3-6t^2+9t$ ,

we obtain

$v(t)=3t^2-12t+9=3(t-1)(t-3)$ .

The particle is instantaneously at rest when $v(t)=0$ , so $t=1$ or $t=3$ . Acceleration is $a(t)=6t-12$ .

Be careful with the phrase distance travelled. Displacement can decrease, but distance cannot, so a distance calculation may need to be split at times when velocity changes sign.

Implicit differentiation at HL

For an equation such as

$x^2+xy+y^2=7$ ,

differentiate every term with respect to $x$ :

$2x+y+x(dy/dx)+2y(dy/dx)=0$ .

Collecting the derivative terms gives

$(x+2y)(dy/dx)=-(2x+y)$ ,

$dy/dx=-(2x+y)/(x+2y)$ .

The essential principle is that differentiating a function of $y$ with respect to $x$ introduces a factor of $dy/dx$ . HL students can target this method through RevisionDojo's implicit differentiation resources.

Showing working that earns marks

IB command terms tell you how much evidence is expected. According to the official guide, differentiate means obtain the derivative, find requires an answer with relevant stages of working, and hence directs you to use the preceding result.

Follow these presentation principles:

Write the derivative before substituting numerical values.
Keep exact values unless an approximation is requested or appropriate.
In a “show that” question, begin from the given information and derive the stated result without assuming it.
When asked to justify, provide mathematical evidence such as a sign change, second derivative, or endpoint comparison.
Include coordinates, intervals, units, and contextual language where required.
Avoid replacing all working with an unexplained calculator result.

On technology-allowed papers, a calculator can solve derivative equations or display graphs, but you still need to communicate the mathematical setup. On Paper 1, fluency with factorization, exact values, and algebraic simplification is especially important. The Paper 1 preparation guide explains how to develop these non-calculator skills.

Why worked video solutions improve differentiation faster

Differentiation is procedural, so improvement depends on seeing exactly where a procedure succeeds or fails. The fastest practical study cycle is:

Attempt one question under timed conditions.
Mark the point where you became uncertain.
Watch the complete per-question worked solution.
Compare the structure and notation with your own method.
Close the solution and reproduce it independently.
Attempt a related question two or three days later.

Watching before attempting often creates false confidence because the method looks obvious once it is presented. Attempting first forces you to make a decision, and the video then corrects that decision while it is still fresh.

RevisionDojo's Maths AA calculus questionbank and calculus video solutions support this attempt-watch-redo cycle. Students who need focused rule practice can use the chain, product, and quotient rule questionbank before moving to mixed questions.

Common differentiation mistakes

The recurring traps are highly predictable:

Missing the chain rule: The derivative of $(3x+1)^5$ is $15(3x+1)^4$ , not .

Keep an error log organized by mistake type rather than by question number. Jojo AI can help explain why a method failed, but you should still redo the calculation independently to confirm that the correction has become part of your own method.

Final exam checklist

Before leaving an IB Maths AA differentiation question, ask:

Have I answered the final command term?
Did I select the correct rule for the function's structure?
Is every chain-rule factor present?
Did I use the original function for coordinates?
Have I justified a maximum, minimum, or inflexion properly?
Did I apply domain restrictions and include units?
Is my final answer exact or rounded appropriately?

Differentiation questions become manageable when you recognize their structure and separate the derivative calculation from its application. Use the RevisionDojo IB Maths AA hub to review the underlying method, then prioritize active practice through the Questionbank and per-question video solutions. Attempting, reviewing, and reattempting worked differentiation questions is more effective than repeatedly reading the same notes.

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