IB · Mathematics: Analysis & Approaches SL & HL · Calculus

Differentiation: IB Mathematics: Analysis & Approaches questions with answers

Free Differentiation questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Calculus. Each question has the answer and the marking points an examiner would look for.

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Differentiation practice questions

Question 1Structured · 4 marks

Let g(x) = (3x2 − 1)3.

Find the gradient of the graph of g at the point where x = 1.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
72
attempt to use the chain rule M1
g′(x) = 3(3x2 − 1)2 × 6x A1A1
g′(1) = 72 A1
4
Question 2Structured · 6 marks

Let f(x) = x2e−x.

(a)

Find f′(x).

[3]
(b)

Find the x-coordinates of the points on the graph of f where the gradient is zero.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)xe−x(−x + 2)
attempt to use the product rule M1
2xe−x − x2e−x A1A1
3
(b)x = 0, x = 2
f′(x) = 0 M1
e−x ≠ 0 R1
x = 0, x = 2 A1
3
Question 3Structured · 4 marks

Let g(x) = (3x2 + 1)5.

Find the gradient of the graph of g at the point where x = 1.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
7680
attempt to use the chain rule M1
g′(x) = 5(3x2 + 1)4 × 6x A1A1
g′(1) = 7680 A1
4
Question 4Structured · 5 marks

Let f(x) = ln xx, x > 0.

(a)

Show that f′(x) = 1 − ln xx2.

[3]
(b)

Find the exact x-coordinate of the maximum point of the graph of f.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
attempt to use the quotient rule M1
x·[[1x − ln x|x2]] A1
simplifying to 1 − ln xx2 AG A1
3
(b)x = e
ln x = 11 (M1)
x = e A1
2
Question 5Structured · 4 marks

Let g(x) = (3x2 − 1)5.

Find the gradient of the graph of g at the point where x = 2.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
878460
attempt to use the chain rule M1
g′(x) = 5(3x2 − 1)4 × 6x A1A1
g′(2) = 878460 A1
4
Question 6Structured · 5 marks

Let f(x) = ln xx3, x > 0.

(a)

Show that f′(x) = 1 − 3ln xx4.

[3]
(b)

Find the exact x-coordinate of the maximum point of the graph of f.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
attempt to use the quotient rule M1
x3·[[1x − 3x2ln x|x6]] A1
simplifying to 1 − 3ln xx4 AG A1
3
(b)x = e1/3
ln x = 13 (M1)
x = e1/3 A1
2

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in