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IB Maths AI SL · Unit 5: Calculus

IB Maths AI SL Differentiation Rules Questions

Exam-style IB Maths AI SL differentiation rules 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.

Practise Differentiation Rules questions → AI SL formula booklet

What you need to know

SL AI differentiation is limited to power rule, sum/difference, and the interpretation of dy/dx as a rate. Always link the derivative to the real-world quantity in the question. Basic differentiation and rates of change overview →

What's examined in AI SL differentiation rules

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

Key formulas

Derivative of x^n
\(f(x) = x^n \implies f'(x) = n x^{\,n-1}\)
Trapezoidal rule (approx. area)
\(A \approx \tfrac{h}{2}\bigl(y_0 + y_n + 2(y_1 + y_2 + \cdots + y_{n-1})\bigr),\ h = \dfrac{b-a}{n}\)

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

Differentiation Rules worked examples

Worked example 1: Power rule and evaluating $f'(x)$ · easy

$f(x) = 4x^3 - 5x^2 + 2x - 7$. Find $f'(x)$ and the exact gradient at $x = 2$.

Solution

1. Power rule: multiply coefficient by the power, drop the power by 1.

2. Differentiate: $f'(x) = 12x^2 - 10x + 2$.

3. Substitute: $f'(2) = 12(4) - 20 + 2$.

4. Evaluate: $48 - 20 + 2 = \mathbf{30}$.

Examiner tip: The derivative of a constant (here $-7$) is zero — a horizontal line has no gradient. The derivative of $2x$ is simply $2$.

Worked example 2: Finding $x$ given a specific gradient · medium

$g(x) = 2x^3 - 3x^2 - 12x$. Find the exact values of $x$ where the gradient equals $24$.

Solution

1. Derivative: $g'(x) = 6x^2 - 6x - 12$.

2. Set equal to $24$: $6x^2 - 6x - 12 = 24$.

3. Rearrange: $6x^2 - 6x - 36 = 0$.

4. Simplify: $x^2 - x - 6 = 0$.

5. Factorise: $(x-3)(x+2) = 0$.

6. State: $\mathbf{x = 3 \text{ or } x = -2}$.

Examiner tip: "Gradient equals $k$" means set the DERIVATIVE equal to $k$, not the original function.

Worked example 3: Finding constants from a point and a gradient · hard

Curve $y = ax^2 + bx$ passes through $(1,5)$ and has gradient $14$ at $x=2$. Find $a$ and $b$ exactly.

Solution

1. Point equation: substitute $(1,5)$ into the curve $\Rightarrow a + b = 5$.

2. Derivative: $\tfrac{dy}{dx} = 2ax + b$.

3. Gradient equation: at $x=2$, $\tfrac{dy}{dx}=14 \Rightarrow 4a + b = 14$.

4. Subtract: $(4a+b) - (a+b) = 14 - 5 \Rightarrow 3a = 9$.

5. Solve: $a = 3$.

6. Back-substitute: $b = 5 - 3 = 2$. So $\mathbf{a=3, b=2}$.

Examiner tip: A point tells you about $f(x)$; a gradient tells you about $f'(x)$. Keep the two substitutions separate when building simultaneous equations.

Try these IB Maths AI SL differentiation rules 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 · 3 marks · Paper 1

Consider the polynomial function \(f(x) = 4x^3 - 5x^2 + 2x - 7\).

  1. Find the derivative function, \(f'(x)\).

  2. Calculate the exact value of \(f'(2)\).

Attempt it and see the mark scheme →

Question 2 · medium · 4 marks · Paper 1

Consider the function \(h(x) = \frac{3x^4 - 2x^2}{x}\) for \(x \neq 0\).

  1. Rewrite \(h(x)\) as a polynomial by dividing each term in the numerator by \(x\).

  2. Find \(h'(x)\).

  3. Determine the value of \(x\) for which the gradient of \(h(x)\) is \(23\).

Attempt it and see the mark scheme →

Question 3 · hard · 5 marks · Paper 1

A cubic function has a derivative given by \(f'(x) = 6x^2 - 10x + c\), where \(c\) is a constant.

  1. Given that the gradient of the curve at \(x = 2\) is \(8\), find the value of \(c\).

  2. Using your value of \(c\), find the \(x\)-coordinates where the gradient of the curve is zero.

Attempt it and see the mark scheme →

All 27 differentiation rules questions with mark schemes →

FAQ

How many IB Maths AI SL differentiation rules questions are there?

There are 27 exam-style differentiation rules questions in the AI SL question bank (Paper 1: 22 · Paper 2: 5), graded 8 easy, 5 medium, 3 hard, 3 very hard, 8 starter. Every question has a full IB-style mark scheme (M, A and R marks).

Is differentiation rules on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 22 · Paper 2: 5. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL 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.

More AI SL Unit 5 topics

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