Q. Determine the derivative of f(x) = x^3 - 4x + 7. (2023)
-
A.
3x^2 - 4
-
B.
3x^2 + 4
-
C.
x^2 - 4
-
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)
-
A.
5x^4 - 9x^2 + 2
-
B.
5x^4 - 9x + 2
-
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)
-
A.
4e^x + 4x^2e^x
-
B.
4x^2e^x + 4xe^x
-
C.
4e^x + 2x^2e^x
-
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) = 4x^2 + 3x - 5. (2019)
-
A.
8x + 3
-
B.
4x + 3
-
C.
2x + 3
-
D.
8x - 3
Solution
Using the power rule, f'(x) = 8x + 3.
Correct Answer:
A
— 8x + 3
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Q. Differentiate f(x) = 4x^5 - 2x^3 + x. (2022)
-
A.
20x^4 - 6x^2 + 1
-
B.
20x^4 - 6x^2
-
C.
4x^4 - 2x^2 + 1
-
D.
5x^4 - 2x^2
Solution
Using the power rule, f'(x) = 20x^4 - 6x^2 + 1.
Correct Answer:
A
— 20x^4 - 6x^2 + 1
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Q. Differentiate f(x) = ln(x^2 + 1). (2022)
-
A.
2x/(x^2 + 1)
-
B.
1/(x^2 + 1)
-
C.
2x/(x^2 - 1)
-
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)
-
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
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 f(x) = x^2 * ln(x).
-
A.
2x * ln(x) + x
-
B.
x * ln(x) + 2x
-
C.
2x * ln(x)
-
D.
x^2/x
Solution
Using the product rule, f'(x) = 2x * ln(x) + x.
Correct Answer:
A
— 2x * ln(x) + x
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Q. Differentiate the function f(x) = ln(x^2 + 1).
-
A.
2x/(x^2 + 1)
-
B.
2/(x^2 + 1)
-
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.
-
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) = 4x^3 - 2x + 1. (2022)
-
A.
12x^2 - 2
-
B.
12x^2 + 2
-
C.
4x^2 - 2
-
D.
4x^2 + 2
Solution
Using the power rule, f'(x) = 12x^2 - 2.
Correct Answer:
A
— 12x^2 - 2
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Q. Find the derivative of f(x) = 5x^2 + 3x - 1. (2020)
-
A.
10x + 3
-
B.
5x + 3
-
C.
10x - 1
-
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^2 + 3x - 7. (2020)
-
A.
10x + 3
-
B.
5x + 3
-
C.
10x - 3
-
D.
5x - 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)
-
A.
15x^2 - 4
-
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)
-
A.
3x^2 * ln(x) + x^2
-
B.
3x^2 * ln(x) + x^3/x
-
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 + 2x^3 - x + 1. (2023)
-
A.
4x^3 + 6x^2 - 1
-
B.
4x^3 + 2x^2 - 1
-
C.
3x^3 + 6x^2 - 1
-
D.
4x^3 + 2x - 1
Solution
Using the power rule, f'(x) = 4x^3 + 6x^2 - 1.
Correct Answer:
A
— 4x^3 + 6x^2 - 1
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Q. Find the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 2.
-
A.
4x^3 - 12x^2 + 12x
-
B.
4x^3 - 12x + 6
-
C.
12x^2 - 4x + 6
-
D.
4x^3 - 12x^2 + 2
Solution
Using the power rule, f'(x) = 4x^3 - 12x^2 + 12x.
Correct Answer:
A
— 4x^3 - 12x^2 + 12x
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Q. Find the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 24x + 5. (2023)
-
A.
4x^3 - 12x^2 + 12x - 24
-
B.
4x^3 - 12x^2 + 6x - 24
-
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 f(x) = x^5 - 2x^3 + x. (2019)
-
A.
5x^4 - 6x^2 + 1
-
B.
5x^4 - 6x
-
C.
5x^4 + 2x^2 + 1
-
D.
5x^4 - 2x^2
Solution
Using the power rule, f'(x) = 5x^4 - 6x^2 + 1.
Correct Answer:
A
— 5x^4 - 6x^2 + 1
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Q. Find the derivative of g(x) = sin(x) + cos(x). (2020)
-
A.
cos(x) - sin(x)
-
B.
-sin(x) - cos(x)
-
C.
sin(x) + cos(x)
-
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)?
-
A.
24x - 4
-
B.
12x - 2
-
C.
12x - 4
-
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) = 5x^2 + 3x - 1, what is f''(x)? (2020)
Solution
The first derivative f'(x) = 10x + 3, and the second derivative f''(x) = 10.
Correct Answer:
A
— 10
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Q. If f(x) = 5x^2 + 3x - 1, what is f'(2)? (2020)
Solution
First, find f'(x) = 10x + 3. Then, f'(2) = 10(2) + 3 = 20 + 3 = 23.
Correct Answer:
A
— 27
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Q. If f(x) = 5x^2 - 3x + 7, what is f''(x)? (2020)
Solution
The first derivative f'(x) = 10x - 3, and the second derivative f''(x) = 10.
Correct Answer:
A
— 10
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Q. If f(x) = x^2 * e^x, find f'(x). (2019)
-
A.
e^x(x^2 + 2x)
-
B.
e^x(x^2 - 2x)
-
C.
x^2 * e^x
-
D.
2x * e^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 * e^x, what is f'(x)? (2019)
-
A.
e^x(x^2 + 2x)
-
B.
e^x(x^2 - 2x)
-
C.
2xe^x
-
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 * ln(x), what is f'(x)? (2022)
-
A.
2x * ln(x) + x
-
B.
x * ln(x) + 2x
-
C.
2x * ln(x) - x
-
D.
x * ln(x) - 2x
Solution
Using the product rule, f'(x) = 2x * ln(x) + x.
Correct Answer:
A
— 2x * ln(x) + x
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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^3 - 4x + 1, what is f''(x)? (2023)
-
A.
6x - 4
-
B.
6x + 4
-
C.
3x^2 - 4
-
D.
3x^2 + 4
Solution
First derivative f'(x) = 3x^2 - 4, then f''(x) = 6x.
Correct Answer:
A
— 6x - 4
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