Chain Rule
Q. If y = (2x + 1)^3, find dy/dx at x = 1.
Solution
dy/dx = 6(2x + 1)^2. At x = 1, dy/dx = 6(2(1) + 1)^2 = 6(3)^2 = 54.
Correct Answer: A — 12
Q. If y = (2x + 1)^3, find dy/dx.
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A.
6(2x + 1)^2
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B.
3(2x + 1)^2
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C.
2(2x + 1)^2
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D.
12(2x + 1)^2
Solution
dy/dx = 6(2x + 1)^2 by the chain rule.
Correct Answer: A — 6(2x + 1)^2
Q. If y = (x^2 + 1)^5, find dy/dx at x = 1.
Solution
dy/dx = 5(x^2 + 1)^4(2x). At x = 1, dy/dx = 5(2)^4(2) = 5(16)(2) = 160.
Correct Answer: B — 20
Q. If y = (x^2 + 1)^5, find dy/dx.
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A.
10x(x^2 + 1)^4
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B.
5(x^2 + 1)^4
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C.
5x(x^2 + 1)^4
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D.
20x(x^2 + 1)^3
Solution
dy/dx = 10x(x^2 + 1)^4 by the chain rule.
Correct Answer: A — 10x(x^2 + 1)^4
Q. If y = 2^x, find dy/dx at x = 1.
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A.
0.693
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B.
1.386
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C.
2.718
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D.
3.141
Solution
dy/dx = 2^x * ln(2). At x = 1, dy/dx = 2^1 * ln(2) = 2 * 0.693 = 1.386.
Correct Answer: A — 0.693
Q. If y = 3x^2 + 2x, find dy/dx at x = 2.
Solution
First, find dy/dx = 6x + 2. At x = 2, dy/dx = 6(2) + 2 = 12 + 2 = 14.
Correct Answer: B — 18
Q. If y = 4x^2 + 3x + 2, find dy/dx at x = -1.
Solution
dy/dx = 8x + 3. At x = -1, dy/dx = 8(-1) + 3 = -8 + 3 = -5.
Correct Answer: A — -5
Q. If y = 4x^3 - 2x + 1, find dy/dx at x = -1.
Solution
dy/dx = 12x^2 - 2. At x = -1, dy/dx = 12(-1)^2 - 2 = 12 - 2 = 10.
Correct Answer: C — -6
Q. If y = 5x^4 + 3x^2 - x, find dy/dx at x = 1.
Solution
dy/dx = 20x^3 + 6x - 1. At x = 1, dy/dx = 20(1)^3 + 6(1) - 1 = 20 + 6 - 1 = 25.
Correct Answer: B — 22
Q. If y = 5x^4 - 3x^3 + 2x - 1, find dy/dx at x = 1.
Solution
dy/dx = 20x^3 - 9x^2 + 2. At x = 1, dy/dx = 20 - 9 + 2 = 13.
Correct Answer: B — 16
Q. If y = cos(5x^2), find dy/dx.
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A.
-10xsin(5x^2)
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B.
-5xsin(5x^2)
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C.
-25xsin(5x^2)
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D.
-2xsin(5x^2)
Solution
dy/dx = -10xsin(5x^2) by the chain rule.
Correct Answer: A — -10xsin(5x^2)
Q. If y = e^(3x), find dy/dx at x = 0.
Solution
dy/dx = 3e^(3x). At x = 0, dy/dx = 3e^(0) = 3.
Correct Answer: A — 1
Q. If y = e^(3x), find dy/dx.
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A.
3e^(3x)
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B.
e^(3x)
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C.
9e^(3x)
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D.
6e^(3x)
Solution
dy/dx = 3e^(3x) by the chain rule.
Correct Answer: A — 3e^(3x)
Q. If y = ln(5x^2 + 3), find dy/dx at x = 1.
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A.
5/8
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B.
3/8
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C.
1/8
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D.
1/5
Solution
dy/dx = (10x)/(5x^2 + 3). At x = 1, dy/dx = (10)/(5(1) + 3) = 10/8 = 5/4.
Correct Answer: A — 5/8
Q. If y = ln(5x^2 + 3), find dy/dx.
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A.
10/(5x^2 + 3)
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B.
5/(5x^2 + 3)
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C.
2/(5x^2 + 3)
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D.
15/(5x^2 + 3)
Solution
dy/dx = (10x)/(5x^2 + 3) by the chain rule.
Correct Answer: A — 10/(5x^2 + 3)
Q. If y = sin(2x), find dy/dx at x = π/4.
Solution
dy/dx = 2cos(2x). At x = π/4, dy/dx = 2cos(π/2) = 2(0) = 0.
Correct Answer: D — √2
Q. If y = sqrt(4x^2 + 1), find dy/dx.
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A.
(4x)/(sqrt(4x^2 + 1))
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B.
(2x)/(sqrt(4x^2 + 1))
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C.
(8x)/(sqrt(4x^2 + 1))
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D.
(2)/(sqrt(4x^2 + 1))
Solution
Using the chain rule, dy/dx = (4x)/(2sqrt(4x^2 + 1)) = (4x)/(sqrt(4x^2 + 1)).
Correct Answer: A — (4x)/(sqrt(4x^2 + 1))
Q. If y = sqrt(x^2 + 1), find dy/dx at x = 0.
Solution
dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x). At x = 0, dy/dx = (1/2)(1)^(-1/2)(0) = 0.
Correct Answer: B — 1
Q. If y = tan(3x), find dy/dx at x = π/6.
Solution
dy/dx = 3sec^2(3x). At x = π/6, dy/dx = 3sec^2(π/2) = 3(∞) = undefined.
Correct Answer: A — 3√3
Q. If y = tan(3x), find dy/dx.
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A.
3sec^2(3x)
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B.
3tan^2(3x)
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C.
sec^2(3x)
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D.
3tan(3x)
Solution
Using the chain rule, dy/dx = 3sec^2(3x).
Correct Answer: A — 3sec^2(3x)
Q. If y = x^3 * e^x, find dy/dx at x = 0.
Solution
Using product rule, dy/dx = 3x^2 * e^x + x^3 * e^x. At x = 0, dy/dx = 0 + 0 = 0.
Correct Answer: A — 0
Q. If y = √(4x^2 + 1), find dy/dx.
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A.
4x/(√(4x^2 + 1))
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B.
2x/(√(4x^2 + 1))
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C.
2/(√(4x^2 + 1))
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D.
8x/(√(4x^2 + 1))
Solution
dy/dx = (4x)/(2√(4x^2 + 1)) = 4x/(√(4x^2 + 1)).
Correct Answer: A — 4x/(√(4x^2 + 1))
Q. If y = √(x^2 + 1), find dy/dx at x = 1.
Solution
dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x). At x = 1, dy/dx = (1/2)(2)^(-1/2)(2) = 1/√2.
Correct Answer: A — 1/√2
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