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. 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
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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 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
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C.
6x^3 - 5x
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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) = 5x^5 - 3x + 7? (2020)
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A.
25x^4 - 3
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B.
15x^4 - 3
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C.
5x^4 - 3
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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)
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B.
2x * cos(x) + x^2 * sin(x)
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C.
2x * sin(x) - x^2 * cos(x)
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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)
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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) + 3x^2
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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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