Water pours into an inverted right circular conical tank at a constant rate of $2\text{ cm}^3\text{s}^{-1}$. The tank has a maximum base radius of $5\text{ cm}$ and a total height of $10\text{ cm}$. Let $h$ be the depth of the water. Find the exact rate at which the water level ($h$) is rising at the instant when the depth of the water is exactly $4\text{ cm}$. (The volume of a cone is $V = \frac{1}{3}\pi r^2 h$)

1. Relate the radius $r$ to the depth $h$ using similar triangles: $\frac{r}{h} = \frac{5}{10} \implies r = 0.5h$. 2. Substitute $r$ into the volume formula to express $V$ solely in terms of $h$: $V = \frac{1}{3}\pi (0.5h)^2 h = \frac{1}{12}\pi h^3$. 3. Differentiate both sides with respect to time $t$ using the chain rule: $\frac{dV}{dt} = \frac{3}{12}\pi h^2 \frac{dh}{dt} = \frac{1}{4}\pi h^2 \frac{dh}{dt}$. 4. Substitute the known rate $\frac{dV}{dt} = 2$ and the specific depth $h = 4$: $2 = \frac{1}{4}\pi (4^2) \frac{dh}{dt}$. 5. Simplify the equation: $2 = 4\pi \frac{dh}{dt}$. 6. Solve exactly for the rate of change of depth: $\frac{dh}{dt} = \frac{1}{2\pi}\text{ cm s}^{-1}$ .

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IB Maths AI HL · Unit 5: Calculus

IB Maths AI HL Advanced Differentiation and Related Rates Questions

Q: How many IB Maths AI HL advanced differentiation and related rates questions are there?

There are 40 exam-style advanced differentiation and related rates questions in the AI HL question bank (Paper 1: 36 · Paper 2: 4), graded 11 easy, 17 medium, 12 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is advanced differentiation and related rates on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 36 · Paper 2: 4. Practise with your GDC — AI papers expect calculator methods throughout.

Q: Where can I get the mark schemes?

Open the AI 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.

Updated October 2026

Exam-style IB Maths AI HL advanced differentiation and related rates questions with worked solutions. Start with the 2 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.

40 questions
Paper 1: 36
Paper 2: 4
11 easy
17 medium
12 hard
2 worked examples

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🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

HL AI extends SL AI with the chain, product and quotient rules and related rates. The 'water tank filling' family of questions turns on d/dt applied to a relationship between variables. Differentiation — chain, product and quotient rules, related rates overview →

HL AI's Paper 3 favourite. Separable DEs by algebra, non-separable by Euler's numerical method. Practise the CG50 keystrokes for Euler. Differential equations — first-order separable and Euler's method overview →

What's examined in AI HL advanced differentiation and related rates

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

Chain, Product, Quotient Rules (26)
Related Rates Problems (13)
Implicit Differentiation & Optimization (1)

Key formulas

Separable differential equation: $\frac{d y}{d x} = f (x) g (y) ⟹ \int \frac{1}{g (y)} d y = \int f (x) d x$
Euler's method: $y_{n + 1} = y_{n} + h f (x_{n}, y_{n}), x_{n + 1} = x_{n} + h$
Trapezoidal rule (refresher): $A \approx \frac{h}{2} (y_{0} + y_{n} + 2 \sum_{i = 1}^{n - 1} y_{i})$

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

Advanced Differentiation and Related Rates worked examples

Worked example 1: Applying the Product Rule · easy

Consider the function $f (x) = x^{2} \ln x$ for $x > 0$ . Find the exact expression for the derivative $f^{'} (x)$ , giving your answer in a fully factorised form.

Solution

5 steps

Examiner tip: When using the product or quotient rule involving natural logarithms, always simplify expressions like

x^{2} \times \frac{1}{x}

immediately to avoid algebraic tangles in subsequent steps.

Worked example 2: Solving a Related Rates Problem · hard

Water pours into an inverted right circular conical tank at a constant rate of $2 {cm}^{3} s^{- 1}$ . The tank has a maximum base radius of $5 cm$ and a total height of $10 cm$ . Let $h$ be the depth of the water. Find the exact rate at which the water level ( $h$ ) is rising at the instant when the depth of the water is exactly $4 cm$ . (The volume of a cone is $V = \frac{1}{3} π r^{2} h$ )

Solution

6 steps

Examiner tip: In 3D related rate problems (like cones), you must substitute the geometrical ratio (e.g.,

r = 0.5 h

) to reduce the volume formula to a single variable before you attempt to differentiate.

Try these IB Maths AI HL advanced differentiation and related rates 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 · 4 marks · Paper 1

Differentiate $y = (3 x^{2} - 5 x)^{4}$ with respect to $x$ . Express your answer in a fully factorised form.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Consider the function $y = \ln (\cos x)$ .
(a) Find $\frac{d y}{d x}$ . [3 marks]
(b) State the continuous domain of values for $x$ containing $x = 0$ for which this derivative is valid. [2 marks]

Full worked solution Free, no sign-in

(a) Use the Chain Rule. The derivative of $\ln (f (x))$ is $\frac{f^{'} (x)}{f (x)}$ . M1
Inner derivative: $\frac{d}{d x} (\cos x) = - \sin x$ . A1
$\frac{d y}{d x} = \frac{- \sin x}{\cos x} = - \tan x$ . A1
(b) The function $\ln (\cos x)$ is only real and defined when the argument is strictly positive: $\cos x > 0$ . M1
For the domain containing zero, this occurs when $- \frac{π}{2} < x < \frac{π}{2}$ . A1

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

Question 3 · hard · 7 marks · Paper 1

Consider the curve given by the equation $y = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}$ .
Use the quotient rule to differentiate this expression and prove algebraically that: $\frac{d y}{d x} = 1 - y^{2}$

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI HL advanced differentiation and related rates questions are there?

There are 40 exam-style advanced differentiation and related rates questions in the AI HL question bank (Paper 1: 36 · Paper 2: 4), graded 11 easy, 17 medium, 12 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is advanced differentiation and related rates on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 36 · Paper 2: 4. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI 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

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Free ready-to-teach lessons (slides and teacher notes): Chain rule (basic composition) · Related rates of change.

IB Maths AI HL Advanced Differentiation and Related Rates Questions

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What you need to know

What's examined in AI HL advanced differentiation and related rates

Key formulas

Advanced Differentiation and Related Rates worked examples

Worked example 1: Applying the Product Rule · easy

Solution

Worked example 2: Solving a Related Rates Problem · hard

Solution

Try these IB Maths AI HL advanced differentiation and related rates questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 1

Revise advanced differentiation and related rates for the exam

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