Q1. If f(x) = c then f'(x) =
(A) c
(B) 0
(C) 1
(D) 2
Q2. d/dx (2x^2 + 3)^5 =
(A) (2x^2 + 3)^4
(B) 5(2x^2 + 3)^4(4)
(C) 5(2x^2 + 3)^4(3)
(D) 5(2x^2 + 3)^4(20x)
Q3. d/dx g(x) =
(A) f(x)g'(x) – f'(x)g(x) / [g(x)]
(B) f'(x)g(x) – f(x)g'(x) / [g'(x)]
(C) g(x)f'(x) – f(x)g'(x) / [g(x)]^2
(D) f'(x)g(x) – f(x)g'(x) / [g(x)]^2
Q4. If x = a (y – a cos y), dy/dx =
(A) cos x
(B) tan x
(C) –cot x
(D) –tan x
Q5. 1 + 1/x^2 is derivatives of:
(A) sin x
(B) sec x
(C) tan x
(D) cot x
Q6. d/dx (e^cos x) equals:
(A) –sin x e^cos x
(B) sin x cos x
(C) cos sin x
(D) None
Q7. Derivatives of ln(1 – cos x) w.r.t x equal:
(A) sin x
(B) tan x / 2
(C) cot x / 2
(D) cot x
Q8. If e^2x =
(A) 16e^2x
(B) 8e^x
(C) 2e^2x
(D) e^2x
Q9. The function f(x) = 3x has minimum value at:
(A) x = 3
(B) x = 1
(C) x = –1
(D) x = 0
Q10. If y = 1/x^2, then dy/dx at x = –1 is:
(A) 2
(B) 3
(C) 1/3
(D) 4
Q11. d/dx (ax + b / 1 + b) equals:
(A) d/dx ax + b
(B) d/dx a / ax + b
(C) d/dx a – ax + b^2
(D) lim(ax + b)
Q12. d/dx (sin x)^2 is equal to:
(A) cos x^2
(B) –cos x^2
(C) 3x^2 cos x^3
(D) x^2 sin x^4
Q13. If f(x) = cos x then f'(f'(f'(f'(f'(x))))) =
(A) 1
(B) 0
(C) –1
(D) 1/2
Q14. If f(x) = 2^x then f'(x) is equals:
(A) 2^x – 1
(B) 2^x ln 2
(C) 2^x
(D) ln 2 / 2
Q15. d/dx (ln 2x) equals:
(A) e^sin x, cos x
(B) –e^sin x, cos x
(C) –e^cos x / sin x
(D) e^cos x, sin x
Q16. d/dx (ln 2x)^2 =
(A) 1/x
(B) 2x
(C) 2/x
(D) 1/x
Q17. If f(x) = h – 2x^h then (2) equals:
(A) ln x
(B) 2 ln 2
(C) 2x ln 2
(D) 2^x
Q18. If f(x) = cos(x – h) then f'(x) equals:
(A) –cos x
(B) cos x
(C) sin x
(D) –sin x
Q19. If f(x) = 1/3 x^3 then f'(f'(2)) is:
(A) 5/8
(B) 1/4
(C) –3/8
Q20. d/dx (x^2 + 1)^2 =
(A) 2(x^2 + 1)
(B) (x^2 + 1)
(C) x(x^2 + 1)
(D) 4x(x^2 + 1)
Q21. x d/dx (sin x)^2 =
(A) 2x cos x^2
(B) cos x^2
(C) x cos x^2
(D) 2x cos x sin x
Q22. d/dx (cos x)^2 =
(A) 2x sin x^2
(B) –2x sin x
(C) x sin x^2
(D) None
Q23. d/dx (a^(f(x))) =
(A) f'(x) a^(f(x)) ln a
(B) f’^(x) a^(f(x))
(C) a^(f(x)) ln a
(D) f'(x) a^(f(x)) / ln a
Q24. d/dx (ln (1/x^2)) =
(A) x
(B) –x
(C) 1/x
(D) –1/x
Q25. If f(x) = sin x then f'(0) =
(A) 1
(B) –1
(C) 0
Q26. f(x) = sin x is decreasing function in the interval:
(A) (π/2, π)
(B) (–π/2, π)
(C) (–π, 0)
(D) (–3π/2, 2π)
Q27. d/dx (tan^–1 x) is equals:
(A) 1 / 1 + x
(B) 1 / 1 – x
(C) 1 / 1 + x^2
(D) 1 / x^2 – 1
Q28. d/dx (ln f(x)) =
(A) 1/x
(B) 1/f(x)
(C) f'(x) / f(x)
(D) f(x).f(x)
Q29. d/dx (–cot x) =
(A) sec^2 x
(B) cosec^2 x
(C) –cosec^2 x
(D) –sec^2 x
Q30. d/dx (x^3 + 4)^3 is equal to:
(A) x (x^3 + 4)^3
(B) x (x^3 + 4)^3 (2x^2)
(C) 2x^2 (x^3 + 4)^3
(D) x (x^3 + 4)
Q31. If f(x) = x f(x) = x^2 then f^–1 (x) =
(A) 2 / x^3
(B) 2 / x^3
(C) 3 / x^3
(D) 3 / 2^3
Q32. lim f(x) – f(a) / x – a =
(A) f^–1 (x)
(B) f^–1 (a)
(C) f^–1 (0)
(D) f^–1 (2)
Q33. d/dx (√(x – 1/x^2)) =
(A) 1 / √x – 1
(B) x + 1 / x
(C) √x – x^2
(D) 1 – 1/x^2
Q34. d/dx (sin 2x) is equal to:
(A) cos 2x
(B) 2 cos 2x
(C) sin x
(D) 2 sin x
Q35. d/dx (cos^–1 x) =
(A) 1 / 1 – x^2
(B) 1 / 1 + x^2
(C) –1 / √(1 – x^2)
(D) √(1 – x^2)
Q36. ln (x + √(x^2 – 1)) =
(A) cosh^–1 x
(B) sinh^–1 x
(C) tanh^–1 x
(D) sech^–1 x
Q37. The critical value of f(x) = x^2 – x equals:
(A) 1/2
(B) 1/2
(C) 2
(D) –2
Q38. A function f^1 > 0 then it is:
(A) Decreasing
(B) Increasing
(C) Decreasing
(D) Zero
Q39. d/dx (cos^–1 x) is equal to:
(A) 1 / √(1 – x^2)
(B) 1 / √(1 + x^2)
(C) –1 / √(1 – x^2)
(D) 1 / √(1 – x^2)
Q40. d/dx (a^x) =
(A) 1 / 1 – x^2
(B) 1 / √(1 – x^2)
(C) –1 / 1 + x^2
(D) 1 / 1 + x^2
Q41. d/dx (a^x ln a) =
(A) (a^x) ln a
(B) a^(x) ln a
(C) A and B
(D) None of these
Q42. ∀ x ∈ (a, b) function f(x) is said to be increasing in (a,b):
(A) f(x) > 0
(B) f(x) < 0
(C) f(x) > 0
(D) f(x) = 0
Q43. d/dx (x^n) is equal to:
(A) nx^(n–1)
(B) nx^(n–1)
(C) A and B
(D) None of these
Q44. f(x) = 1 / x – 2 =
(A) 1
(B) 0
(C) 2
(D) ∞
Q45. d/dx (tan x) =
(A) –cosh x
(B) tan^2 x
(C) sec^2 x
(D) None of these
Q46. d/dx (ln (e^x + e^–x)) is equal to:
(A) e^x – e^–x / e^x + e^–x
(B) e^x – e^–x / e^x + e^–x
(C) e^x + e^–x / e^x – e^–x
(D) None of these
Q47. d/dx (sinh x) is equal to:
(A) cosh x
(B) cosh x
(C) tanh x
(D) sech x
Q48. MacLaurin expression of ln (1 + x) is:
(A) x – x^2 / 2! + x^3 / 3! + …
(B) x^2 – x^3 / 2! + x^4 / 3! + …
(C) x^2 – x^2 / 2! + x^4 / 4! + …
(D) x – x^2 / 2! + x^3 / 3! + …
Q49. d/dx (log a x) =
(A) x / ln a
(B) ln a
(C) 1 / x
(D) 1 / x ln a
Q50. If y = cos x u = sin x then dy/du =
(A) cos x d/dx
(B) –cot x
(C) –tan x
(D) tan x
Q51. d/dx (a^x) =
(A) x / a
(B) 1 / a y with respect to x is given by x / a
(C) x / a
(D) a
Q52. d/dx (x^3) is equal to:
(A) x^3
(B) x^2
(C) 3x^2
(D) x
Q53. d/dx (5 f(x)) =
(A) 5 f^–1 (x)
(B) 5
(C) 5 f'(x)
(D) (5 + f^–1 (x))
Q54. If y = sec x (3π/2 – x) then y equals:
(A) cosec x cot x (π/2 – x)
(B) –cosec x cot x
(C) sec x tan x
(D) –sec x tan x
Q55. d/dx (e^√x) =
(A) e^√x
(B) –e^√x / √x
(C) e^√x / √x
(D) –e^√x / 2√x
Q56. d/dx (sech x) is equal to:
(A) sech x tan x
(B) –sech x tan x
(C) sec x tanh x
(D) sech x tanh x
Q57. f(x) = f(0) + f^–1 x + f^–1 x / 2! + …
(A) Taylor series
(B) Binomial series
(C) Laurent series
(D) Maclaurin series
Q58. d/dx (√x) =
(A) 1 / 2x√x
(B) –1 / 2x√x
(C) –1 / 2x
(D) None of these
Q60. d/dx (x^a) where a is a constant:
(A) 1/x
(B) –a/x
(C) a/x
(D) x
Q61. d/dx (tan^–1 x) =
(A) 1 / x √(x^2 – 1)
(B) 1 / 1 + x^2
(C) –1 / 1 + x^2
(D) cot^–1 x
Q62. d/dx (e^tan x) =
(A) e^tan x
(B) e^tan x ln x sec^2 x
(C) e^tan x ln x tan x
(D) e^tan x sec^2 x
Q63. d/dx (log_10 (x + 1)) =
(A) 1 + x
(B) 1 / (1 + x) ln 10
(C) –1 / (1 + x) ln 10
(D) 1 / 1 + x
Q64. If y = sin 3x then y^2 =
(A) 3cos 3x
(B) 9cos 3x
(C) –9cos 3x
(D) sin 3x
Q65. Let “C” be differential function in neighborhood of “C” where f has relative maximum at:
(A) f'(C) = 0
(B) f'(C) > 0
(C) f'(C) < 0
(D) f”(C) = 0
Q66. The derivative of √x at x = a is:
(A) 1 / 2√a
(B) 1 / √a
(C) √a / 2
(D) 2√a
Q67. d/dx (ax + b)^n is equal to:
(A) n(ax + b)^n
(B) n(ax + b)^n–1
(C) n ax^–1
(D) n a (ax + b)^n–1
Q68. (fog)'(x) =
(A) f'(g(x)) g'(x)
(B) f(g(x))
(C) f'(g(x)) g'(x)
(D) None of these
Q69. d/dx (tan^–1 x – cot^–1 x) =
(A) 1 / 1 + x^2
(B) 1 / 1 + x^2
(C) –1 / 1 – x^2
(D) 1 / 1 – x^2
Q70. If f(x) = cosh x then if f(x) = cosh x then (f'(x))^2 – (f(x))^2 =
(A) 0
(B) 1
(C) 2
(D) 4
Q71. d/dx (e^(x+h)) =
(A) ln h
(B) ln x
(C) a^(x+h) ln x
(D) h a^(x+h)
Q72. If √(cot x) = y then dy/dx =
(A) cosec^2 x
(B) –cosec^2 x
(C) 2√(cot x)
(D) –2√(cot x)
Q73. Derivative of (x^3 + 1)^3 w.r.t x^2:
(A) (x^3 + 1)^8
(B) 9(x^3 + 1)^9
(C) 27x (x^3 + 1)^8
(D) 27x (x^3 + 1)^9
Q74. If y = √x then dy/dx =
(A) 2√x
(B) 1 / 2√x
(C) √x
(D) 2 / √x
Q75. d/dx (sin x cos 2x) is equal to:
(A) sin x cos x
(B) (sin x cos x)^2
(C) sin x cos x
(D) cos x
Q76. d/dx (cot^–1 x) =
(A) 1 / 1 + x^2
(B) –1 / 1 + x^2
(C) 1 / 1 – x^2
(D) –1 / 1 – x^2
Q77. d/dx (e^(f(x))) is equal to:
(A) e^(f(x))
(B) e^(f(x)) f'(x)
(C) f'(x) / e^x
(D) e^(x) / e^x
Q78. d/dx (e^ln x^2) =
(A) (e^ln x^2)
(B) (e^2 ln x^2)
(C) A and B
(D) 2x