Differentiation
Q. Calculate the derivative of f(x) = 5x^5. (2016)
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A.
25x^4
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B.
5x^4
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C.
20x^4
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D.
10x^4
Solution
The derivative f'(x) = d/dx(5x^5) = 25x^4.
Correct Answer: A — 25x^4
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Q. Calculate the derivative of f(x) = ln(x^2 + 1).
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A.
2x/(x^2 + 1)
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B.
1/(x^2 + 1)
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C.
2/(x^2 + 1)
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D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023
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A.
e^x(2x + 1)
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B.
e^x(2x - 1)
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C.
2xe^x
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D.
x^2e^x
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x(2x + 1).
Correct Answer: A — e^x(2x + 1)
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Q. Find the derivative of f(x) = sin(x) + cos(x).
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A.
cos(x) - sin(x)
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B.
-sin(x) - cos(x)
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C.
sin(x) + cos(x)
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D.
-cos(x) + sin(x)
Solution
The derivative f'(x) = d/dx(sin(x) + cos(x)) = cos(x) - sin(x).
Correct Answer: A — cos(x) - sin(x)
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Q. Find the derivative of f(x) = tan(x). (2022) 2022
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A.
sec^2(x)
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B.
csc^2(x)
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C.
sec(x)
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D.
tan^2(x)
Solution
The derivative f'(x) = d/dx(tan(x)) = sec^2(x).
Correct Answer: A — sec^2(x)
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Q. Find the derivative of f(x) = x^5 + 3x^3 - 2x.
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A.
5x^4 + 9x^2 - 2
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B.
5x^4 + 6x^2 - 2
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C.
3x^2 + 5x^4 - 2
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D.
5x^4 + 3x^2 - 2
Solution
The derivative f'(x) = d/dx(x^5 + 3x^3 - 2x) = 5x^4 + 9x^2 - 2.
Correct Answer: A — 5x^4 + 9x^2 - 2
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Q. Find the derivative of f(x) = x^5 - 3x^3 + 2. (2022)
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A.
5x^4 - 9x^2
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B.
5x^4 + 9x^2
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C.
3x^2 - 9x
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D.
5x^4 - 3x^2
Solution
The derivative f'(x) = d/dx(x^5 - 3x^3 + 2) = 5x^4 - 9x^2.
Correct Answer: A — 5x^4 - 9x^2
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Q. What is the derivative of f(x) = 1/x?
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A.
-1/x^2
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B.
1/x^2
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C.
-x
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D.
x
Solution
The derivative f'(x) = d/dx(1/x) = -1/x^2.
Correct Answer: A — -1/x^2
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Q. What is the derivative of f(x) = 3x^2 + 5x - 7? (2021) 2021
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A.
3x + 5
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B.
6x + 5
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C.
6x - 5
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D.
3x^2 + 5
Solution
The derivative f'(x) = d/dx(3x^2 + 5x - 7) = 6x + 5.
Correct Answer: B — 6x + 5
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Q. What is the derivative of f(x) = 5x^4?
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A.
20x^3
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B.
5x^3
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C.
15x^3
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D.
4x^3
Solution
The derivative f'(x) = d/dx(5x^4) = 20x^3.
Correct Answer: A — 20x^3
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Q. What is the derivative of f(x) = e^(2x)?
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A.
2e^(2x)
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B.
e^(2x)
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C.
e^(x)
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D.
2x*e^(2x)
Solution
Using the chain rule, f'(x) = d/dx(e^(2x)) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. What is the derivative of f(x) = x^3 - 4x + 6?
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A.
3x^2 - 4
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B.
3x^2 + 4
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C.
x^2 - 4
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D.
3x^2 - 6
Solution
The derivative f'(x) = d/dx(x^3 - 4x + 6) = 3x^2 - 4.
Correct Answer: A — 3x^2 - 4
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Q. What is the derivative of f(x) = x^3 - 4x?
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A.
3x^2 - 4
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B.
3x^2 + 4
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C.
x^2 - 4
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D.
3x^2 - 2
Solution
The derivative f'(x) = d/dx(x^3 - 4x) = 3x^2 - 4.
Correct Answer: A — 3x^2 - 4
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Q. What is the derivative of f(x) = x^4 + 2x^3 - x + 1? (2017)
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A.
4x^3 + 6x^2 - 1
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B.
4x^3 + 2x^2 - 1
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C.
3x^3 + 6x^2 - 1
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D.
4x^3 + 2x - 1
Solution
The derivative f'(x) = d/dx(x^4 + 2x^3 - x + 1) = 4x^3 + 6x^2 - 1.
Correct Answer: A — 4x^3 + 6x^2 - 1
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