IB Maths AA HL · Unit 5: Calculus
IB Maths AA HL Advanced Differentiation Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 34 questions
- Paper 1: 25
- Paper 2: 9
- 7 easy
- 9 medium
- 13 hard
- 5 starter
- 3 worked examples
Practise Advanced Differentiation questions →
AA HL formula booklet
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 (12)
- Implicit Differentiation (12)
- Related Rates (10)
Key formulas
- Euler's identity
- \(e^{i\pi} + 1 = 0\)
In the same notation as the IB formula booklet. All AA HL formulas →
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 $\frac{dy}{dx}$ in terms of $x$ and $y$.
1. Differentiate both sides of the equation with respect to $x$.
2. Apply the standard power rule to $x^2$ to get $2x$.
3. Apply the chain rule for implicit differentiation to $y^2$ to get $2y \frac{dy}{dx}$.
4. Set the differentiated expression equal to $0$ (since the derivative of the constant $25$ is $0$): $2x + 2y \frac{dy}{dx} = 0$.
5. Rearrange to isolate the derivative term: $2y \frac{dy}{dx} = -2x$.
6. Divide to solve for the final gradient function: $\mathbf{\frac{dy}{dx} = -\frac{x}{y}}$.
Examiner tip: When performing implicit differentiation, a very common error is forgetting to multiply by $\frac{dy}{dx}$ 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 $y = \arccos(2x)$ at the point where $x = \frac{1}{4}$.
1. Identify the standard derivative formula for $\arccos(u)$ from the formula booklet: $\frac{d}{dx}(\arccos u) = -\frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$.
2. Apply the chain rule by letting $u = 2x$, so $\frac{du}{dx} = 2$.
3. Substitute into the formula to find the gradient function: $\frac{dy}{dx} = -\frac{2}{\sqrt{1 - (2x)^2}} = -\frac{2}{\sqrt{1 - 4x^2}}$.
4. Evaluate the tangent gradient at $x = \frac{1}{4}$: $m_t = -\frac{2}{\sqrt{1 - 4(\frac{1}{16})}} = -\frac{2}{\sqrt{1 - \frac{1}{4}}} = -\frac{2}{\sqrt{3/4}}$.
5. Simplify the tangent gradient: $m_t = -\frac{2}{\sqrt{3}/2} = -\frac{4}{\sqrt{3}}$.
6. Calculate the normal gradient, which is the negative reciprocal $m_n = -\frac{1}{m_t}$: $\mathbf{m_n = \frac{\sqrt{3}}{4}}$.
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 $10 \text{ cm}^3\text{s}^{-1}$. 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: $\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt} = 8\pi(5) \times \frac{1}{10\pi} = 40\pi \times \frac{1}{10\pi} = \mathbf{4 \text{ cm}^2\text{s}^{-1}}$.
Examiner tip: Set up your related rates chain rules explicitly (e.g., $\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}$) 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.
Question 1 · easy · 2 marks · Paper 1
Find the exact derivative of the function \(f(x) = 5^x\) with respect to \(x\).
Question 2 · medium · 5 marks · Paper 2
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 \(288\pi \text{ cm}^3\).
Question 3 · hard · 6 marks · Paper 1
By rewriting \(y = \arcsin x\) as \(\sin y = x\), use implicit differentiation to formally prove that:
\[\frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}\]
All 34 advanced differentiation questions with mark schemes →
FAQ
How many IB Maths AA HL advanced differentiation questions are there?
There are 34 exam-style advanced differentiation questions in the AA HL question bank (Paper 1: 25 · Paper 2: 9), graded 7 easy, 9 medium, 13 hard, 5 starter. 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: 25 · Paper 2: 9. 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.
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