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Q. Find the critical points of the function f(x) = x^3 - 6x^2 + 9x.

A. (0, 0)
B. (3, 0)
C. (2, 0)
D. (1, 0)

Solution

f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Critical points are (1, f(1)) and (3, f(3)).

Correct Answer: B — (3, 0)

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Q. Find the critical points of the function f(x) = x^4 - 8x^2 + 16. (2019)

A. (0, 16)
B. (2, 0)
C. (4, 0)
D. (1, 9)

Solution

Setting f'(x) = 4x^3 - 16x = 0 gives x = 0, ±2. Evaluating f(2) = 0 shows (2, 0) is a critical point.

Correct Answer: B — (2, 0)

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Q. Find the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6).

A. (-3, 6, -3)
B. (0, 0, 0)
C. (3, -6, 3)
D. (1, -2, 1)

Solution

Cross product A × B = |i  j  k| |1  2  3| |4  5  6| = (-3, 6, -3).

Correct Answer: A — (-3, 6, -3)

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Q. Find the derivative of f(x) = 1/x.

A. -1/x^2
B. 1/x^2
C. -2/x^2
D. 1/x

Solution

Using the power rule, f'(x) = -1/x^2.

Correct Answer: A — -1/x^2

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Q. Find the derivative of f(x) = 3x^2 + 5x - 7.

A. 6x + 5
B. 3x + 5
C. 6x - 5
D. 3x^2 + 5

Solution

Using the power rule, f'(x) = d/dx(3x^2) + d/dx(5x) - d/dx(7) = 6x + 5.

Correct Answer: A — 6x + 5

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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^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) = 5x^4 - 3x + 2.

A. 20x^3 - 3
B. 15x^3 - 3
C. 20x^4 - 3
D. 5x^3 - 3

Solution

Using the power rule, f'(x) = 20x^3 - 3.

Correct Answer: A — 20x^3 - 3

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Q. Find the derivative of f(x) = e^(2x) at x = 0.

A. 1
B. 2
C. e
D. 2e

Solution

f'(x) = 2e^(2x). At x = 0, f'(0) = 2e^0 = 2.

Correct Answer: B — 2

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Q. Find the derivative of f(x) = e^(2x).

A. 2e^(2x)
B. e^(2x)
C. 2xe^(2x)
D. e^(x)

Solution

Using the chain rule, f'(x) = 2e^(2x).

Correct Answer: A — 2e^(2x)

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Q. Find the derivative of f(x) = e^(x^2).

A. 2xe^(x^2)
B. e^(x^2)
C. x e^(x^2)
D. 2e^(x^2)

Solution

Using the chain rule, f'(x) = e^(x^2) * 2x = 2x e^(x^2).

Correct Answer: A — 2xe^(x^2)

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Q. Find the derivative of f(x) = e^x * ln(x) at x = 1.

A. 1
B. 0
C. e
D. ln(e)

Solution

Using the product rule, f'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.

Correct Answer: A — 1

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Q. Find the derivative of f(x) = e^x * sin(x) at x = 0.

A. 1
B. 0
C. e
D. sin(0)

Solution

Using the product rule, f'(0) = e^0 * sin(0) + e^0 * cos(0) = 0 + 1 = 1.

Correct Answer: A — 1

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Q. Find the derivative of f(x) = ln(x^2 + 1) at x = 1.

A. 0
B. 1
C. 1/2
D. 1/3

Solution

f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 2/2 = 1.

Correct Answer: B — 1

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Q. Find the derivative of f(x) = ln(x^2 + 1).

A. 2x/(x^2 + 1)
B. 1/(x^2 + 1)
C. 2/(x^2 + 1)
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. Find the derivative of f(x) = sin(x) + cos(x) at x = π/4.

A. 0
B. 1
C. √2
D. √2/2

Solution

f'(x) = cos(x) - sin(x), thus f'(π/4) = √2/2 - √2/2 = 0.

Correct Answer: C — √2

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Q. Find the derivative of f(x) = sin(x) + cos(x).

A. cos(x) - sin(x)
B. -sin(x) - cos(x)
C. sin(x) + cos(x)
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) = sin(x) at x = π/2.

A. 0
B. 1
C. -1
D. undefined

Solution

f'(x) = cos(x); f'(π/2) = cos(π/2) = 0.

Correct Answer: B — 1

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Q. Find the derivative of f(x) = tan(x) at x = 0.

A. 0
B. 1
C. undefined
D. 1/2

Solution

f'(x) = sec^2(x); f'(0) = sec^2(0) = 1.

Correct Answer: B — 1

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Q. Find the derivative of f(x) = tan(x) at x = π/4.

A. 1
B. 2
C. √2
D. 0

Solution

f'(x) = sec^2(x). At x = π/4, f'(π/4) = sec^2(π/4) = 2.

Correct Answer: A — 1

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Q. Find the derivative of f(x) = tan(x).

A. sec^2(x)
B. csc^2(x)
C. sin^2(x)
D. cos^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) = tan(x). (2022) 2022

A. sec^2(x)
B. csc^2(x)
C. sec(x)
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^2 * e^x.

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) = x^2 * e^x + 2x * e^x = e^x(x^2 + 2x).

Correct Answer: A — e^x(x^2 + 2x)

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Q. Find the derivative of f(x) = x^2 sin(1/x) at x = 0.

A. 0
B. 1
C. undefined
D. does not exist

Solution

Using the limit definition of the derivative, we find that f'(0) = 0, hence it is differentiable at x = 0.

Correct Answer: A — 0

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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^3 - 3x^2 + 4 at x = 2.

A. 6
B. 8
C. 10
D. 12

Solution

f'(x) = 3x^2 - 6x. At x = 2, f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.

Correct Answer: B — 8

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Q. Find the derivative of f(x) = x^3 - 3x^2 + 4x - 5.

A. 3x^2 - 6x + 4
B. 3x^2 - 3x + 4
C. 3x^2 - 6x + 5
D. 3x^2 + 6x - 4

Solution

Using the power rule, f'(x) = 3x^2 - 6x + 4.

Correct Answer: A — 3x^2 - 6x + 4

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Q. Find the derivative of f(x) = x^3 - 4x^2 + 6x.

A. 3x^2 - 8x + 6
B. 3x^2 - 4x + 6
C. 3x^2 - 8x
D. x^2 - 4x + 6

Solution

Using the power rule, f'(x) = 3x^2 - 8x + 6.

Correct Answer: A — 3x^2 - 8x + 6

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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 + 3x^3 - 2x.

A. 5x^4 + 9x^2 - 2
B. 5x^4 + 6x^2 - 2
C. 3x^2 + 5x^4 - 2
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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Major Competitive Exams

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