Pure Mathematics - Differentiation Practice Questions
A-Levels · A-Level Mathematics · 147 free MCQs with instant results and detailed explanations.
147
Total
33
Easy
76
Medium
38
Hard
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Sample Questions from Pure Mathematics - Differentiation
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Q1
Easy
If f(x) = x^3 - 6x^2 + 9x, what is f'(2)?
A.
3
B.
0
C.
6
D.
9
Show Answer & Explanation
Correct Answer: B
First, we find the derivative f'(x) = 3x^2 - 12x + 9. Evaluating at x=2, we have f'(2) = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3, so check computation, and indeed f'(2) = 0.
Q2
Easy
The function g(x) = 2sin(x) + 4cos(x) is differentiated. What is g'(x)?
A.
2cos(x) - 4sin(x)
B.
-2sin(x) + 4cos(x)
C.
4sin(x) - 2cos(x)
D.
-2cos(x) - 4sin(x)
Show Answer & Explanation
Correct Answer: A
Using the derivative rules for sine and cosine, g'(x) = 2cos(x) - 4sin(x). The derivative of sin(x) is cos(x) and for cos(x) it is -sin(x), so we apply these rules.
Q3
Easy
If f(x) = x^3 - 6x^2 + 9x, what is f'(2)?
A.
3
B.
6
C.
0
D.
-3
Show Answer & Explanation
Correct Answer: C
To find f'(2), first, we calculate the derivative f'(x) = 3x^2 - 12x + 9. Substituting x = 2 into the derivative gives f'(2) = 3(2^2) - 12(2) + 9 = 0.
Q4
Medium
What is the derivative of f(x) = 3x^4 - 5x^3 + 2x - 7?
A.
12x^3 - 15x^2 + 2
B.
12x^3 - 15x^2 + 1
C.
12x^3 - 5x^2 + 2
D.
3x^3 - 5x^2 + 2
Show Answer & Explanation
Correct Answer: A
The derivative is calculated by applying the power rule. Each term's exponent is multiplied by its coefficient and then reduced by one. Thus, the derivative is 12x^3 - 15x^2 + 2.
Q5
Medium
If f(x) = x^3 + 2x^2 - 4x, what is f'(2)?
A.
10
B.
12
C.
8
D.
6
Show Answer & Explanation
Correct Answer: A
First, find the derivative f'(x) = 3x^2 + 4x - 4. Then, substitute x = 2 into the derivative: f'(2) = 3(2^2) + 4(2) - 4 = 12 + 8 - 4 = 10.
Q6
Medium
The function f(x) = e^(2x)sin(x) is differentiated. What is the correct form of f'(x)?
A.
e^(2x)(2sin(x) + cos(x))
B.
e^(2x)(sin(x) + 2cos(x))
C.
2e^(2x)sin(x) + e^(2x)cos(x)
D.
2e^(2x)cos(x) + e^(2x)sin(x)
Show Answer & Explanation
Correct Answer: A
Using the product rule for differentiation: f'(x) = u'v + uv', where u = e^(2x) and v = sin(x). Thus, f'(x) = (2e^(2x))sin(x) + e^(2x)cos(x). Hence, the derivative simplifies to e^(2x)(2sin(x) + cos(x)).
Q7
Medium
If g(x) = ln(3x^2 + 1), what is g'(x)?
A.
6x/(3x^2 + 1)
B.
3x/(3x^2 + 1)
C.
2x/(3x^2 + 1)
D.
1/(3x^2 + 1)
Show Answer & Explanation
Correct Answer: A
Using the chain rule, g'(x) = (1/(3x^2 + 1))(6x), which simplifies to 6x/(3x^2 + 1). Here, the derivative of the inner function (3x^2 + 1) is 6x.
Q8
Hard
Given the function f(x) = 3x^4 - 8x^3 + 6x^2 - 5, what is the value of x at which the function has a local maximum?
A.
1
B.
2
C.
0
D.
3
Show Answer & Explanation
Correct Answer: B
To find local maxima, we set the first derivative f'(x) = 0 and test for concavity using the second derivative. The calculations show that x = 2 is a local maximum.
Q9
Hard
If the function g(x) = e^(2x) * sin(x), find g''(0).
A.
0
B.
2
C.
1
D.
e^0
Show Answer & Explanation
Correct Answer: C
To find g''(0), we differentiate g(x) twice. The second derivative at x = 0 yields g''(0) = 1.
Q10
Hard
If f(x) = 3x^4 - 8x^3 + 5x^2 - 2, what is f''(1)?
A.
34
B.
56
C.
18
D.
40
Show Answer & Explanation
Correct Answer: A
To find f''(1), we first differentiate f(x) twice. The first derivative f'(x) is 12x^3 - 24x^2 + 10x. The second derivative f''(x) is 36x^2 - 48x + 10. Evaluating f''(1) gives 36(1)^2 - 48(1) + 10 = 34.
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Pure Mathematics - Differentiation — A-Levels A-Level Mathematics Practice Questions Online
This page contains 147 practice MCQs for the chapter Pure Mathematics - Differentiation in A-Levels A-Level Mathematics. The questions are organized by difficulty — 33 easy, 76 medium, 38 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.