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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Differentiation (Set 3)

Take the ECAT Mathematics — Differentiation (Set 3): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. d/dx (x³ cos x) equals:

    • A. 3x² cos x + x³ sin x
    • B. 3x² sin x − x³ cos x
    • C. 3x² cos x − x³ sin x ✓
    • D. −3x² sin x

    Answer: C. 3x² cos x − x³ sin x

    Explanation: Product rule: (x³)′cos x + x³(cos x)′ = 3x² cos x − x³ sin x.

  2. Q2. Where sin x > 0, the derivative of √(sin x) with respect to x is:

    • A. 1/(2√(sin x))
    • B. cos x / √(sin x)
    • C. −cos x / (2√(sin x))
    • D. cos x / (2√(sin x)) ✓

    Answer: D. cos x / (2√(sin x))

    Explanation: Chain rule: (1/(2√u))·u′ with u = sin x gives cos x/(2√(sin x)).

  3. Q3. d/dx (eˣ sin x) equals:

    • A. eˣ(sin x + cos x) ✓
    • B. eˣ cos x
    • C. eˣ(sin x − cos x)
    • D. −eˣ cos x

    Answer: A. eˣ(sin x + cos x)

    Explanation: Product rule: eˣ sin x + eˣ cos x.

  4. Q4. d/dx [x/(1 + x²)] equals:

    • A. 1/(2x)
    • B. (1 − x²)/(1 + x²)² ✓
    • C. (1 + x²)/(1 + x²)²
    • D. (x² − 1)/(1 + x²)²

    Answer: B. (1 − x²)/(1 + x²)²

    Explanation: Quotient rule: [(1 + x²)·1 − x·2x]/(1 + x²)² = (1 − x²)/(1 + x²)².

  5. Q5. For −1 < x < 1, d/dx ln[(1 + x)/(1 − x)] equals:

    • A. 2/(1 + x²)
    • B. 1/(1 − x²)
    • C. 2/(1 − x²) ✓
    • D. −2/(1 − x²)

    Answer: C. 2/(1 − x²)

    Explanation: ln[(1 + x)/(1 − x)] = ln(1 + x) − ln(1 − x), whose derivative is 1/(1 + x) + 1/(1 − x) = 2/(1 − x²).

The other 25 questions

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  1. Q6. d/dx [sin(ln x)] for x > 0 equals:

    • A. sin(1/x)
    • B. cos(ln x)
    • C. cos(ln x)/x
    • D. −cos(ln x)/x

  2. Q7. d/dx ln(sec x + tan x) equals:

    • A. sec x tan x
    • B. sec x
    • C. 1/(sec x + tan x)
    • D. tan x

  3. Q8. d/dx sinh⁻¹x equals:

    • A. 1/√(1 − x²)
    • B. 1/√(x² + 1)
    • C. 1/(1 + x²)
    • D. 1/√(x² − 1)

  4. Q9. d/dx (5^(x²)) equals:

    • A. 2x·5^(x²)
    • B. x²·5^(x² − 1)
    • C. 5^(x²)·ln 5
    • D. 2x·5^(x²)·ln 5

  5. Q10. If f(x) = eˣ cos x, then f′(0) equals:

    • A. 0
    • B. 2
    • C. 1
    • D. −1

  6. Q11. If y = ln x, then d²y/dx² at x = 2 equals:

    • A. −1/4
    • B. −1/2
    • C. 1/4
    • D. 1/2

  7. Q12. If y = sin 2x, then d²y/dx² at x = π/12 equals:

    • A. 2
    • B. −1
    • C. −2
    • D. −4

  8. Q13. The values of x for which dy/dx = 0 when y = x⁴ − 2x² are:

    • A. x = 0, ½, −½
    • B. x = 1, −1 only
    • C. x = 0, 1, −1
    • D. x = 0, 2, −2

  9. Q14. If eʸ = x + y, then dy/dx equals:

    • A. eʸ/(1 − eʸ)
    • B. 1/(eʸ − 1)
    • C. 1/eʸ
    • D. 1/(eʸ + 1)

  10. Q15. If x = t² and y = t³ (t ≠ 0), then d²y/dx² equals:

    • A. 3/(2t)
    • B. 3/(4t)
    • C. 3t/2
    • D. 3/(4t²)

  11. Q16. If x = a(θ − sin θ) and y = a(1 − cos θ), then dy/dx equals:

    • A. cos θ/(1 − cos θ)
    • B. cot(θ/2)
    • C. tan(θ/2)
    • D. −cot(θ/2)

  12. Q17. The fourth derivative of 1/x is:

    • A. 6/x⁴
    • B. 24/x⁴
    • C. −24/x⁵
    • D. 24/x⁵

  13. Q18. The sixth derivative of x⁵ + 3x² is:

    • A. 6
    • B. 120
    • C. 0
    • D. 720

  14. Q19. If y = a e^(2x) + b e^(−2x), where a and b are constants, which relation holds?

    • A. dy/dx = 2y
    • B. d²y/dx² = 2y
    • C. d²y/dx² = 4y
    • D. d²y/dx² = −4y

  15. Q20. From first principles, lim (h→0) [√(x + h) − √x]/h (x > 0) equals:

    • A. 2√x
    • B. 1/√x
    • C. 1/(2√x)
    • D. √x/2

  16. Q21. The value of lim (h→0) [cos(π/3 + h) − cos(π/3)]/h is:

    • A. −1/2
    • B. −√3/2
    • C. √3/2
    • D. 1/2

  17. Q22. Which statement is true for f(x) = |x| at x = 0?

    • A. f is continuous but not differentiable
    • B. f is neither continuous nor differentiable
    • C. f is differentiable with f′(0) = 1
    • D. f is differentiable with f′(0) = 0

  18. Q23. d/dx (x tan⁻¹x) equals:

    • A. tan⁻¹x + x/(1 + x²)
    • B. tan⁻¹x − x/(1 + x²)
    • C. 1/(1 + x²)
    • D. tan⁻¹x + 1/(1 + x²)

  19. Q24. d/dx ln√(x² + 1) equals:

    • A. 1/(x² + 1)
    • B. 2x/(x² + 1)
    • C. 1/(2√(x² + 1))
    • D. x/(x² + 1)

  20. Q25. If y = x^(2x) for x > 0, then dy/dx at x = 1 equals:

    • A. 2
    • B. 0
    • C. 1
    • D. 2e

  21. Q26. If y = (x + 1)(x + 2)(x + 3), then dy/dx at x = 0 equals:

    • A. 9
    • B. 12
    • C. 6
    • D. 11

  22. Q27. d/dx [(x² + 1)/(x² − 1)] equals:

    • A. −2x/(x² − 1)²
    • B. 4x/(x² − 1)²
    • C. 1
    • D. −4x/(x² − 1)²

  23. Q28. d/dx cos(√x), for x > 0, equals:

    • A. −sin(1/(2√x))
    • B. −sin(√x)
    • C. −sin(√x)/(2√x)
    • D. sin(√x)/(2√x)

  24. Q29. If y = tan x, then d²y/dx² equals:

    • A. 2 sec x tan x
    • B. −2 sec²x tan x
    • C. 2 sec²x tan x
    • D. sec²x

  25. Q30. If y = x sin x, then d²y/dx² equals:

    • A. 2 cos x − x sin x
    • B. cos x − x sin x
    • C. −x sin x
    • D. 2 cos x + x sin x