A spherical balloon is being inflated such that its volume $V$ is increasing at a constant rate of $10 \text{ cm}^3\text{s}^{-1}$. Find the exact rate of change of its surface area $A$ when the radius is exactly $5 \text{ cm}$.

1. Identify the given rate $\frac{dV}{dt} = 10$, and state the formulas for a sphere: $V = \frac{4}{3}\pi r^3$ and $A = 4\pi r^2$. 2. Differentiate the volume with respect to radius: $\frac{dV}{dr} = 4\pi r^2$. 3. Apply the chain rule to link volume and radius rates: $\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt} \implies 10 = 4\pi(5^2) \times \frac{dr}{dt}$. 4. Solve for the rate of change of the radius: $\frac{dr}{dt} = \frac{10}{100\pi} = \frac{1}{10\pi}$. 5. Differentiate the surface area with respect to radius: $\frac{dA}{dr} = 8\pi r$. 6. Apply the chain rule again to find the requested rate: $\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt} = 8\pi(5) \times \frac{1}{10\pi} = 40\pi \times \frac{1}{10\pi} = \mathbf{4 \text{ cm}^2\text{s}^{-1}}$.

A spherical balloon is being inflated such that its volume increases at a constant rate of $15 \text{ cm}^3\text{ s}^{-1}$ . Calculate the rate of increase of its radius at the instant when the volume of the balloon is $288\pi \text{ cm}^3$ .

Find the radius when $V = 288\pi$ : $\frac{4}{3}\pi r^3 = 288\pi \Rightarrow r^3 = 216$ (M1) $r = 6 \text{ cm}$ A1 Differentiate $V = \frac{4}{3}\pi r^3$ : $\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}$ M1 Substitute: $15 = 4\pi(6)^2\frac{dr}{dt} = 144\pi\frac{dr}{dt}$ A1 $\frac{dr}{dt} = \frac{15}{144\pi} = \frac{5}{48\pi} \approx 0.0332 \text{ cm s}^{-1}$ A1

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IB Maths AA HL · Unit 5: Calculus

IB Maths AA HL Advanced Differentiation Questions

Updated October 2026

Exam-style IB Maths AA HL advanced differentiation questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

43 questions
Paper 1: 33
Paper 2: 10
9 easy
17 medium
17 hard
3 worked examples

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What you need to know

HL AA covers first-order separable DEs and linear DEs solved by integrating factor. Paper 3 often layers a Maclaurin series check on top. Differential equations — separable and integrating factor overview →

HL AA extends SL AA's toolkit with implicit differentiation, logarithmic differentiation (for y = x^x-style functions), and derivatives of arcsin, arccos, arctan. Differentiation — implicit, logarithmic, and inverse-trig overview →

What's examined in AA HL advanced differentiation

The question bank covers these advanced differentiation question types (number of questions in brackets):

Advanced Function Derivatives (17)
Implicit Differentiation (13)
Related Rates (13)

Key formulas

Euler's identity: $e^{i π} + 1 = 0$
Derivative of tan x: $\frac{d}{d x} (\tan x) = \sec^{2} x$

In the same notation as the IB formula booklet. All AA HL formulas →

Explained, with a worked example: Chain rule · Product rule · Quotient rule

Advanced Differentiation worked examples

Worked example 1: Implicit differentiation basics · easy

A circle is defined by the implicit equation $x^{2} + y^{2} = 25$ . Find an expression for the gradient function $\frac{d y}{d x}$ in terms of $x$ and $y$ .

Solution

6 steps

Examiner tip: When performing implicit differentiation, a very common error is forgetting to multiply by

\frac{d y}{d x}

when differentiating terms containing

y

. Think of it as a mandatory chain rule step.

Worked example 2: Inverse trigonometric derivatives · medium

Find the exact gradient of the normal to the curve $y = \arccos (2 x)$ at the point where $x = \frac{1}{4}$ .

Solution

6 steps

Examiner tip: For inverse trigonometric derivatives, forgetting to multiply the entire fraction by the derivative of the inner argument (the chain rule step) guarantees the loss of all subsequent accuracy marks.

Worked example 3: Related rates of change · hard

A spherical balloon is being inflated such that its volume $V$ is increasing at a constant rate of $10 {cm}^{3} s^{- 1}$ . Find the exact rate of change of its surface area $A$ when the radius is exactly $5 cm$ .

Solution

6 steps

Examiner tip: Set up your related rates chain rules explicitly (e.g.,

\frac{d A}{d t} = \frac{d A}{d r} \times \frac{d r}{d t}

) before substituting any numbers. Substituting constants into geometric formulas before differentiating will erroneously yield a derivative of zero!

Try these IB Maths AA HL advanced differentiation questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 2 marks · Paper 1

Find the exact derivative of the function $f (x) = 5^{x}$ with respect to $x$ .

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 2

A spherical balloon is being inflated such that its volume increases at a constant rate of $15 {cm}^{3} s^{- 1}$ . Calculate the rate of increase of its radius at the instant when the volume of the balloon is $288 π {cm}^{3}$ .

Full worked solution Free, no sign-in

Find the radius when $V = 288 π$ : $\frac{4}{3} π r^{3} = 288 π \Rightarrow r^{3} = 216$ (M1)
$r = 6 cm$ A1
Differentiate $V = \frac{4}{3} π r^{3}$ : $\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$ M1
Substitute: $15 = 4 π (6)^{2} \frac{d r}{d t} = 144 π \frac{d r}{d t}$ A1
$\frac{d r}{d t} = \frac{15}{144 π} = \frac{5}{48 π} \approx 0.0332 {cm s}^{- 1}$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 1

By rewriting $y = \arcsin x$ as $\sin y = x$ , use implicit differentiation to formally prove that: $\frac{d y}{d x} = \frac{1}{\sqrt{1 - x^{2}}}$

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AA HL advanced differentiation questions are there?

There are 43 exam-style advanced differentiation questions in the AA HL question bank (Paper 1: 33 · Paper 2: 10), graded 9 easy, 17 medium, 17 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is advanced differentiation on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 33 · Paper 2: 10. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA HL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

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Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Implicit differentiation · Related rates of change · Derivatives of trig, inverse trig, exponential & log functions.

IB Maths AA HL Advanced Differentiation Questions

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What you need to know

What's examined in AA HL advanced differentiation

Key formulas

Advanced Differentiation worked examples

Worked example 1: Implicit differentiation basics · easy

Solution

Worked example 2: Inverse trigonometric derivatives · medium

Solution

Worked example 3: Related rates of change · hard

Solution

Try these IB Maths AA HL advanced differentiation questions

Question 1 · easy · 2 marks · Paper 1

Question 2 · medium · 5 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 1

Revise advanced differentiation for the exam

FAQ

What else is free here

Teaching this topic?

More AA HL Unit 5 topics