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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Differentiation (Set 4)

Take the ECAT Mathematics — Differentiation (Set 4): 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 [ln(sin 2x)] equals

    • A. 2 cot 2x ✓
    • B. cos 2x/sin x
    • C. cot 2x
    • D. 2 tan 2x

    Answer: A. 2 cot 2x

    Explanation: By the chain rule: (1/sin 2x)·cos 2x·2 = 2 cot 2x.

  2. Q2. d/dx [tan⁻¹(3x)] equals

    • A. 3/(1 + 9x²) ✓
    • B. 1/(3(1 + x²))
    • C. 1/(1 + 9x²)
    • D. 3/(1 + 3x²)

    Answer: A. 3/(1 + 9x²)

    Explanation: d/dx tan⁻¹u = u′/(1 + u²) with u = 3x, giving 3/(1 + 9x²).

  3. Q3. d/dx [√(x² + 1)] equals

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

    Answer: A. x/√(x² + 1)

    Explanation: (1/2)(x² + 1)^(−1/2) · 2x = x/√(x² + 1).

  4. Q4. d/dx (cos³x) equals

    • A. 3 cos²x sin x
    • B. −3 sin²x cos x
    • C. −3 cos²x sin x ✓
    • D. 3 cos²x

    Answer: C. −3 cos²x sin x

    Explanation: Chain rule: 3cos²x · (−sin x) = −3cos²x sin x.

  5. Q5. d/dx (cosh 2x) equals

    • A. sinh 2x
    • B. 2 cosh 2x
    • C. −2 sinh 2x
    • D. 2 sinh 2x ✓

    Answer: D. 2 sinh 2x

    Explanation: d/dx cosh u = sinh u · u′, so the derivative is 2 sinh 2x.

The other 25 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

  1. Q6. If y = x^(1/x) (x > 0), then dy/dx equals

    • A. x^(1/x − 2)(1 + ln x)
    • B. x^(1/x)(1 − ln x)
    • C. x^(1/x − 2)(1 − ln x)
    • D. x^(1/x − 1)/x

  2. Q7. The value of d/dx [eˣ sin x] at x = π/2 is

    • A. e^(π/2)·π/2
    • B. e^(π/2)
    • C. 2e^(π/2)
    • D. 0

  3. Q8. d/dx [ln(x + √(x² − 1))] equals

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

  4. Q9. For the curve x³ + y³ = 3xy, the slope dy/dx at (3/2, 3/2) is

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

  5. Q10. If x = t² + 1 and y = t³, then d²y/dx² at t = 2 is

    • A. 3/16
    • B. 3
    • C. 3/4
    • D. 3/8

  6. Q11. If y = e²ˣ sin x, then y″ equals

    • A. e²ˣ(4 sin x + cos x)
    • B. e²ˣ(3 sin x + 4 cos x)
    • C. e²ˣ(3 sin x − 4 cos x)
    • D. e²ˣ(5 sin x + 4 cos x)

  7. Q12. For 0 < x < 1/√2, d/dx [sin⁻¹(2x√(1 − x²))] equals

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

  8. Q13. For |x| < 1, d/dx [tan⁻¹((1 + x)/(1 − x))] equals

    • A. 0
    • B. 1/(1 + x²)
    • C. 2/(1 + x²)
    • D. 1/(1 − x²)

  9. Q14. Using the definition f′(x) = lim(h→0)[f(x + h) − f(x)]/h, the derivative of f(x) = 1/x² is

    • A. 2/x³
    • B. −2/x³
    • C. −1/x⁴
    • D. 1/(2x)

  10. Q15. The number of points at which f(x) = |x − 1| + |x − 3| is not differentiable is

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

  11. Q16. If y = (x² + 1)³(x − 1)²/(x + 2)⁴, then dy/dx at x = 0 is

    • A. −1/16
    • B. −1/4
    • C. 1/4
    • D. −4

  12. Q17. d/dx (x^(ln x)) equals

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

  13. Q18. d/dx [sinh⁻¹(x²)] equals

    • A. 1/√(1 + x⁴)
    • B. 2x/√(1 + x⁴)
    • C. 2x/√(x⁴ − 1)
    • D. x²/√(1 + x⁴)

  14. Q19. d/dx [ln tan(x/2)] equals

    • A. cot x
    • B. sec x
    • C. ½ cosec x
    • D. cosec x

  15. Q20. d/dx (e^(√x)) equals

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

  16. Q21. d⁴/dx⁴ (sin 2x) equals

    • A. 16 cos 2x
    • B. 8 sin 2x
    • C. −16 sin 2x
    • D. 16 sin 2x

  17. Q22. For y = A cos(ln x) + B sin(ln x), x²y″ + xy′ + y equals

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

  18. Q23. If y = e²ˣ cos 3x satisfies y″ − 4y′ + cy = 0, then c equals

    • A. 12
    • B. 5
    • C. −13
    • D. 13

  19. Q24. For y = (sin x)^(cos x), the value of dy/dx at x = π/2 is

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

  20. Q25. For x = a cos³θ, y = a sin³θ, dy/dx at θ = π/6 is

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

  21. Q26. d/dx [tan⁻¹(√x)] equals

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

  22. Q27. If y = x²eˣ, the value of the fifth derivative y⁽⁵⁾ at x = 0 is

    • A. 20
    • B. 10
    • C. 60
    • D. 40

  23. Q28. Let f(x) = x² sin(1/x) for x ≠ 0 and f(0) = 0. Then f′(0) equals

    • A. 1
    • B. does not exist
    • C. −1
    • D. 0

  24. Q29. d/dx [x/√(1 + x²)] equals

    • A. 1/(1 + x²)
    • B. (1 − x²)/(1 + x²)^(3/2)
    • C. (1 + x²)^(−1/2)
    • D. (1 + x²)^(−3/2)

  25. Q30. If y = tanh x, then dy/dx in terms of y is

    • A. 1 − y²
    • B. y² − 1
    • C. 1/(1 + y²)
    • D. 1 + y²