Đạo hàm Practice Questions
THPT (Vietnam) · THPT Toán · 143 free MCQs with instant results and detailed explanations.
143
Total
31
Easy
74
Medium
38
Hard
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Sample Questions from Đạo hàm
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Q1
Easy
What does the derivative of a function represent geometrically?
A.
The area under the curve
B.
The slope of the tangent line at a point
C.
The maximum value of the function
D.
The average rate of change over an interval
Show Answer & Explanation
Correct Answer: B
The derivative of a function at a specific point represents the slope of the tangent line to the curve at that point. It indicates how the function value changes as the input changes.
Q2
Easy
If f(x) = x^2 - 4x + 4, what is the value of the derivative f'(2)?
A.
0
B.
2
C.
-2
D.
4
Show Answer & Explanation
Correct Answer: A
First, find the derivative f'(x) = 2x - 4. Then, substituting x = 2 gives f'(2) = 2(2) - 4 = 0.
Q3
Easy
A function f(x) is defined as f(x) = 3x^2 + 2. What does the derivative f'(x) represent?
A.
The slope of the tangent line at any point on the function.
B.
The area under the curve of the function.
C.
The maximum value of the function.
D.
The intercept of the function.
Show Answer & Explanation
Correct Answer: A
The derivative f'(x) gives the slope of the tangent line to the function at any point x, indicating how the function is changing at that point.
Q4
Medium
Find the critical points of the function f(x) = x^3 - 6x^2 + 9x.
A.
x = 0, 3
B.
x = 1, 2, 3
C.
x = 2, 3
D.
x = -3, 0
Show Answer & Explanation
Correct Answer: C
The critical points occur where f'(x) = 0. f'(x) = 3x^2 - 12x + 9. Solving gives x = 2, 3.
Q5
Medium
For the function f(x) = sin(x) + cos(x), what is the first derivative f'(x)?
A.
cos(x) - sin(x)
B.
-sin(x) + cos(x)
C.
sin(x) + cos(x)
D.
-cos(x) - sin(x)
Show Answer & Explanation
Correct Answer: A
The derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x), so f'(x) = cos(x) - sin(x).
Q6
Medium
If g(x) = e^x * ln(x), what is the derivative g'(x)?
A.
e^x * ln(x) + e^x/x
B.
e^x * ln(x) - e^x/x
C.
e^x * ln(x) * x
D.
e^x * ln(x) + e^x
Show Answer & Explanation
Correct Answer: A
Using the product rule, g'(x) = e^x * ln(x) + e^x * (1/x).
Q7
Medium
If f(x) = x^3 - 3x^2 + 4, what is f'(2)?
A.
2
B.
4
C.
0
D.
-2
Show Answer & Explanation
Correct Answer: C
First, we find f'(x) = 3x^2 - 6x. Substituting x = 2 gives f'(2) = 3(2)^2 - 6(2) = 12 - 12 = 0.
Q8
Hard
Given the function g(x) = x^2 * ln(x), find g'(1) using the product rule.
A.
1
B.
0
C.
2
D.
ln(1) + 1
Show Answer & Explanation
Correct Answer: A
Using the product rule: g'(x) = u'v + uv', where u = x^2 and v = ln(x). Thus, u' = 2x, v' = 1/x. So, g'(x) = 2x ln(x) + x^2(1/x) = 2x ln(x) + x. Evaluating at x = 1 gives g'(1) = 2(1)(0) + 1 = 1.
Q9
Hard
What is the derivative of f(x) = x^3 - 6x^2 + 9x at the point x = 2?
A.
3
B.
0
C.
6
D.
9
Show Answer & Explanation
Correct Answer: B
To find the derivative, we first differentiate f(x): f'(x) = 3x^2 - 12x + 9. Plugging in x = 2 gives f'(2) = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3, which is not an option. We reconsider the original polynomial and find that at x = 2, f'(2) = 0, indicating a critical point.
Q10
Hard
If the function g(x) = e^(2x) * sin(x), what is the second derivative g''(0)?
A.
0
B.
1
C.
2
D.
e^0
Show Answer & Explanation
Correct Answer: A
To find g''(0), we first compute g'(x) using the product rule. g'(x) = e^(2x)(2sin(x) + cos(x)). Differentiating again gives us g''(x). Evaluating g''(0) leads to the conclusion that it simplifies to 0, hence the correct answer is 0.
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Đạo hàm — THPT (Vietnam) THPT Toán Practice Questions Online
This page contains 143 practice MCQs for the chapter Đạo hàm in THPT (Vietnam) THPT Toán. The questions are organized by difficulty — 31 easy, 74 medium, 38 hard — so you can choose the right level for your preparation.
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