Q. Determine the derivative of f(x) = x^3 - 4x + 7. (2023)
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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 - 7
Solution
Using the power rule, f'(x) = 3x^2 - 4.
Correct Answer: A — 3x^2 - 4
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Q. Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)
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A.
5x^4 - 9x^2 + 2
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B.
5x^4 - 9x + 2
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C.
5x^4 - 3x^2 + 2
-
D.
5x^4 - 3x^3
Solution
Using the power rule, f'(x) = 5x^4 - 9x^2 + 2.
Correct Answer: A — 5x^4 - 9x^2 + 2
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Q. Differentiate f(x) = 4x^2 * e^x. (2022)
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A.
4e^x + 4x^2e^x
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B.
4x^2e^x + 4xe^x
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C.
4e^x + 2x^2e^x
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D.
8xe^x
Solution
Using the product rule, f'(x) = 4e^x + 4x^2e^x.
Correct Answer: A — 4e^x + 4x^2e^x
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Q. Differentiate f(x) = ln(x^2 + 1). (2022)
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A.
2x/(x^2 + 1)
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B.
1/(x^2 + 1)
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C.
2x/(x^2 - 1)
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D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = 2x/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. Differentiate f(x) = x^2 * e^x. (2022)
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A.
x^2 * e^x + 2x * e^x
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B.
2x * e^x + x^2 * e^x
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C.
x^2 * e^x + e^x
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D.
2x * e^x
Solution
Using the product rule, f'(x) = x^2 * e^x + 2x * e^x.
Correct Answer: A — x^2 * e^x + 2x * e^x
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Q. Differentiate the function f(x) = ln(x^2 + 1).
-
A.
2x/(x^2 + 1)
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B.
2/(x^2 + 1)
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C.
1/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. Differentiate the function f(x) = x^2 * e^x.
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A.
x^2 * e^x + 2x * e^x
-
B.
2x * e^x + x^2 * e^x
-
C.
x^2 * e^x + e^x
-
D.
2x * e^x + e^x
Solution
Using the product rule, f'(x) = (x^2)' * e^x + x^2 * (e^x)' = 2x * e^x + x^2 * e^x.
Correct Answer: A — x^2 * e^x + 2x * e^x
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Q. Find the derivative of f(x) = 5x^2 + 3x - 1. (2020)
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A.
10x + 3
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B.
5x + 3
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C.
10x - 1
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D.
5x^2 + 3
Solution
Using the power rule, f'(x) = 10x + 3.
Correct Answer: A — 10x + 3
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Q. Find the derivative of f(x) = 5x^3 - 4x + 7. (2019)
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A.
15x^2 - 4
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B.
15x^2 + 4
-
C.
5x^2 - 4
-
D.
5x^2 + 4
Solution
Using the power rule, f'(x) = 15x^2 - 4.
Correct Answer: A — 15x^2 - 4
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Q. Find the derivative of f(x) = x^3 * ln(x). (2023)
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A.
3x^2 * ln(x) + x^2
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B.
3x^2 * ln(x) + x^3/x
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C.
3x^2 * ln(x) + x^3
-
D.
3x^2 * ln(x) + 1
Solution
Using the product rule, f'(x) = (x^3)' * ln(x) + x^3 * (ln(x))' = 3x^2 * ln(x) + x^2.
Correct Answer: A — 3x^2 * ln(x) + x^2
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Q. Find the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 24x + 5. (2023)
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A.
4x^3 - 12x^2 + 12x - 24
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B.
4x^3 - 12x^2 + 6x - 24
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C.
4x^3 - 12x^2 + 12x
-
D.
4x^3 - 12x^2 + 6x
Solution
Using the power rule, f'(x) = 4x^3 - 12x^2 + 12x - 24.
Correct Answer: A — 4x^3 - 12x^2 + 12x - 24
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Q. Find the derivative of g(x) = sin(x) + cos(x). (2020)
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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
Using the derivatives of sine and cosine, g'(x) = cos(x) - sin(x).
Correct Answer: A — cos(x) - sin(x)
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Q. If f(x) = 3x^2 + 2x, what is f'(2)? (2023)
Solution
First, find f'(x) = 6x + 2. Then, f'(2) = 6(2) + 2 = 12 + 2 = 14.
Correct Answer: A — 10
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Q. If f(x) = 4x^3 - 2x^2 + x, what is f''(x)?
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A.
24x - 4
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B.
12x - 2
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C.
12x - 4
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D.
24x - 2
Solution
First, find f'(x) = 12x^2 - 4x + 1, then differentiate again to get f''(x) = 24x - 4.
Correct Answer: A — 24x - 4
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Q. If f(x) = x^2 * e^x, what is f'(x)? (2019)
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A.
e^x(x^2 + 2x)
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B.
e^x(x^2 - 2x)
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C.
2xe^x
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D.
x^2e^x
Solution
Using the product rule, f'(x) = e^x(x^2 + 2x).
Correct Answer: A — e^x(x^2 + 2x)
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Q. If f(x) = x^2 + 3x + 5, what is f''(x)? (2020)
Solution
The first derivative f'(x) = 2x + 3, and the second derivative f''(x) = 2.
Correct Answer: A — 2
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Q. If f(x) = x^4 - 2x^3 + x, what is f'(1)? (2023)
Solution
First, find f'(x) = 4x^3 - 6x^2 + 1. Then, f'(1) = 4(1)^3 - 6(1)^2 + 1 = 4 - 6 + 1 = -1.
Correct Answer: A — 2
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Q. If h(x) = e^(2x), what is h'(x)? (2019)
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A.
2e^(2x)
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B.
e^(2x)
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C.
2xe^(2x)
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D.
e^(x)
Solution
Using the chain rule, h'(x) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. What is the derivative of f(x) = 3x^4 - 5x^2 + 2? (2021)
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A.
12x^3 - 10x
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B.
12x^3 - 5
-
C.
6x^3 - 5x
-
D.
3x^3 - 5
Solution
Using the power rule, f'(x) = 12x^3 - 10x.
Correct Answer: A — 12x^3 - 10x
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Q. What is the derivative of f(x) = 3x^4 - 5x^3 + 2x - 7?
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A.
12x^3 - 15x^2 + 2
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B.
12x^3 - 15x^2 - 2
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C.
3x^3 - 5x^2 + 2
-
D.
3x^3 - 5x^2 - 2
Solution
Using the power rule, f'(x) = 12x^3 - 15x^2 + 2.
Correct Answer: A — 12x^3 - 15x^2 + 2
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Q. What is the derivative of f(x) = 4/x? (2022)
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A.
-4/x^2
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B.
4/x^2
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C.
-4/x
-
D.
4/x
Solution
Using the power rule, f'(x) = -4/x^2.
Correct Answer: A — -4/x^2
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Q. What is the derivative of f(x) = 5x^2 - 4x + 3?
-
A.
10x - 4
-
B.
10x + 4
-
C.
5x - 4
-
D.
5x + 4
Solution
Using the power rule, f'(x) = 10x - 4.
Correct Answer: A — 10x - 4
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Q. What is the derivative of f(x) = 5x^5 - 3x + 7? (2020)
-
A.
25x^4 - 3
-
B.
15x^4 - 3
-
C.
5x^4 - 3
-
D.
20x^4 - 3
Solution
Using the power rule, f'(x) = 25x^4 - 3.
Correct Answer: A — 25x^4 - 3
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Q. What is the derivative of f(x) = x^2 * sin(x)? (2023)
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A.
2x * sin(x) + x^2 * cos(x)
-
B.
2x * cos(x) + x^2 * sin(x)
-
C.
2x * sin(x) - x^2 * cos(x)
-
D.
x^2 * sin(x) + 2x * cos(x)
Solution
Using the product rule, f'(x) = 2x * sin(x) + x^2 * cos(x).
Correct Answer: A — 2x * sin(x) + x^2 * cos(x)
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Q. What is the derivative of f(x) = x^3 * ln(x)? (2023)
-
A.
3x^2 * ln(x) + x^2
-
B.
3x^2 * ln(x) + x^3/x
-
C.
3x^2 * ln(x) + 3x^2
-
D.
3x^2 * ln(x) + 1
Solution
Using the product rule, f'(x) = 3x^2 * ln(x) + x^2.
Correct Answer: A — 3x^2 * ln(x) + x^2
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Q. What is the derivative of f(x) = x^4 - 6x^2 + 9? (2022)
-
A.
4x^3 - 12x
-
B.
4x^3 + 12x
-
C.
2x^3 - 6x
-
D.
2x^3 + 6x
Solution
Using the power rule, f'(x) = 4x^3 - 12x.
Correct Answer: A — 4x^3 - 12x
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