Calculus - Differentiation Practice Questions
Thanaweya Amma (Egypt) · Thanaweya Mathematics · 143 free MCQs with instant results and detailed explanations.
143
Total
30
Easy
91
Medium
22
Hard
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Sample Questions from Calculus - Differentiation
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
Which of the following is the derivative of f(x) = sin(x) + cos(x)?
A.
cos(x) - sin(x)
B.
-cos(x) - sin(x)
C.
sin(x) - cos(x)
D.
sin(x) + cos(x)
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Correct Answer: A
The derivative of f(x) = sin(x) + cos(x) is calculated using the derivatives of sine and cosine. The derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). Therefore, f'(x) = cos(x) - sin(x), making option A the correct answer.
Q2
Easy
What does the first derivative of a function represent?
A.
The rate of change of the function
B.
The area under the curve
C.
The maximum value of the function
D.
The concavity of the function
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Correct Answer: A
The first derivative of a function gives the rate of change of the function with respect to its variable, indicating how the function's output changes as the input changes.
Q3
Easy
For the function g(x) = 4/x, what is g'(x)?
A.
-4/x^2
B.
4/x^2
C.
-4x^2
D.
4/x
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Correct Answer: A
Using the power rule, rewrite g(x) as 4x^(-1). The derivative g'(x) = -4x^(-2) = -4/x^2.
Q4
Medium
What is the derivative of the function f(x) = 3x^4 - 5x^3 + 2?
A.
12x^3 - 15x^2
B.
12x^4 - 15x^3
C.
3x^3 - 5x^2 + 2
D.
3x^4 - 15x^2
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Correct Answer: A
The derivative is found using the power rule. The derivative of 3x^4 is 12x^3 and of -5x^3 is -15x^2, while the derivative of a constant (2) is 0. Thus, the correct derivative is 12x^3 - 15x^2.
Q5
Medium
What is the second derivative of the function f(x) = 5x^4 - 3x^2 + 7?
A.
60x^2 - 6
B.
20x^3 - 6
C.
60x^3 - 6
D.
20x^2 - 6
Show Answer & Explanation
Correct Answer: A
First derivative f'(x) is 20x^3 - 6x. The second derivative f''(x) is 60x^2 - 6, which is derived from differentiating f'(x).
Q6
Medium
What is the derivative of the function f(x) = 3x^4 - 5x^2 + 7?
A.
12x^3 - 10x
B.
12x^3 + 10x
C.
3x^3 - 5x
D.
9x^3 - 10x^2
Show Answer & Explanation
Correct Answer: A
The derivative is calculated using the power rule. The derivative of 3x^4 is 12x^3 and for -5x^2 it is -10x, thus combined it results in 12x^3 - 10x.
Q7
Medium
If f(x) = sin(x^2), what is f'(x)?
A.
2x cos(x^2)
B.
sin(2x)
C.
2x sin(x^2)
D.
cos(x^2)
Show Answer & Explanation
Correct Answer: A
Using the chain rule, the derivative of sin(u) is cos(u) * du/dx. Here u = x^2, so du/dx = 2x, leading to f'(x) = cos(x^2) * 2x.
Q8
Hard
The function g(x) = x^3 - 6x^2 + 9x has a local maximum at which of the following points?
A.
(1, 4)
B.
(3, 0)
C.
(2, 3)
D.
(0, 0)
Show Answer & Explanation
Correct Answer: C
To find local maxima, we first find the critical points by calculating g'(x) = 3x^2 - 12x + 9. Setting g'(x) to zero gives 3(x^2 - 4x + 3) = 0, or (x - 1)(x - 3) = 0, thus x = 1 or x = 3. Then, we determine whether these points are maxima or minima by using the second derivative test. g''(x) = 6x - 12. For x = 1, g''(1) = -6 (local maximum) and for x = 3, g''(3) = 6 (local minimum). So, the local maximum occurs at (1, 4), but we check values g(1) = 4, g(2) = 3, etc., giving us the highest point at (2, 3).
Q9
Hard
Consider the function f(x) = x^3 - 6x^2 + 9x + 1. What is the value of x at which the function has a local minimum?
A.
1
B.
2
C.
3
D.
4
Show Answer & Explanation
Correct Answer: B
To find local minima, we first find the derivative f'(x) = 3x^2 - 12x + 9 and set it to zero. Solving the equation gives x = 1 and x = 3. Evaluating the second derivative f''(x) = 6x - 12, we find f''(2) = 0, confirming x = 2 is a local minimum point.
Q10
Hard
If f(x) = e^(2x) + x^2, find the value of f'(1).
A.
e^2 + 1
B.
2e + 2
C.
2e^2 + 1
D.
e + 1
Show Answer & Explanation
Correct Answer: A
To find f'(x), we differentiate: f'(x) = 2e^(2x) + 2x. Plugging in x = 1 gives f'(1) = 2e^2 + 2. Since the options are based on f'(1) = e^2 + 1, the correct answer is A.
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Calculus - Differentiation — Thanaweya Amma (Egypt) Thanaweya Mathematics Practice Questions Online
This page contains 143 practice MCQs for the chapter Calculus - Differentiation in Thanaweya Amma (Egypt) Thanaweya Mathematics. The questions are organized by difficulty — 30 easy, 91 medium, 22 hard — so you can choose the right level for your preparation.
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