TRB BT Assistant - Higher Order Derivatives and Leibnitz Theorem Online Test (MCQ Quiz)

1. What is the second derivative of a function f(x)?

A) The first derivative of f(x) B) The derivative of the first derivative of f(x) C) The integral of f(x) D) The limit of f(x) as x approaches infinity

2. If y = ax^n, what is the second derivative, d^2y/dx^2?

A) anx^(n-1) B) a n(n-1)x^(n-2) C) an(n-1)x^n D) ax^n

3. What is the third derivative of y = sin(x)?

A) cos(x) B) -sin(x) C) -cos(x) D) sin(x)

4. If f(x) = e^(kx), what is the nth derivative of f(x)?

A) k^n * e^(kx) B) e^(kx) C) k * e^(kx) D) n * e^(kx)

5. Leibnitz Theorem is used to find:

A) The first derivative of a product of two functions B) The second derivative of a product of two functions C) The nth derivative of a product of two functions D) The nth derivative of a single function

6. State the Leibnitz Theorem for the nth derivative of the product of two functions u(x) and v(x).

A) d^n(uv)/dx^n = sum_{r=0}^{n} C(n,r) * (d^r u / dx^r) * (d^(n-r) v / dx^(n-r)) B) d^n(uv)/dx^n = sum_{r=0}^{n} C(n,r) * (d^(n-r) u / dx^(n-r)) * (d^r v / dx^r) C) d^n(uv)/dx^n = sum_{r=0}^{n} C(n,r) * (d^r u / dx^r) + (d^(n-r) v / dx^(n-r)) D) d^n(uv)/dx^n = sum_{r=0}^{n} C(n,r) * (d^r u / dx^r) * (d^r v / dx^r)

7. In the Leibnitz Theorem formula, C(n,r) represents:

A) The r-th derivative of n B) The binomial coefficient 'n choose r' C) The factorial of n divided by r D) The combination of n items taken r at a time

8. If y = x^3, what is the fourth derivative of y?

A) 6x B) 6 C) 0 D) 1

9. What is the value of the second derivative of y = ln(x) at x=1?

A) 1 B) -1 C) 0 D) 2

10. Let u = x^2 and v = e^x. Using Leibnitz Theorem, find the second derivative of y = uv.

A) e^x(x^2 + 4x + 2) B) e^x(x^2 + 2x + 2) C) e^x(x^2 + 4x) D) e^x(x^2 + 2)

11. If y = a^x, what is the nth derivative of y?

A) (ln a)^n * a^x B) a^x C) n * a^(x-1) D) (ln a) * a^x

12. What is the third derivative of y = cos(x)?

A) sin(x) B) cos(x) C) -sin(x) D) -cos(x)

13. Consider the function y = x^m. For what value of m is the (m+1)th derivative equal to zero?

A) Any positive integer m B) m=0 C) m=1 D) m=2

14. If y = x^2 * sin(x), what is the first term in the expansion of the second derivative using Leibnitz Theorem?

A) x^2 * sin(x) B) 2x * sin(x) C) sin(x) D) 2 * sin(x)

15. What is the nth derivative of a constant function?

A) The constant itself B) 0 C) 1 D) Undefined

16. If y = log_a(x), what is the second derivative of y?

A) -1/x^2 B) -1/(ax^2) C) -1/(x^2 * ln a) D) -1/x

17. Leibnitz Theorem is a generalization of which rule?

A) Chain Rule B) Product Rule C) Quotient Rule D) Sum Rule

18. What is the value of the third derivative of y = x^4 at x=2?

A) 24 B) 48 C) 96 D) 192

19. If y = uv, and u = x, v = sin(x), find the third derivative of y using Leibnitz Theorem.

A) x cos(x) - 3 sin(x) B) x cos(x) + 3 sin(x) C) 3x cos(x) - sin(x) D) 3x sin(x) + cos(x)

20. What is the nth derivative of y = x^n?

A) n! B) n C) 1 D) 0

21. If y = tan(x), what is the second derivative at x=0?

A) 0 B) 1 C) 2 D) tan(0)

22. Which of the following functions has a third derivative that is identically zero?

A) x^4 B) x^3 C) x^2 D) x

23. Let y = x^2 * e^x. Find the second derivative using Leibnitz Theorem.

A) e^x(x^2 + 4x + 2) B) e^x(x^2 + 2x + 2) C) e^x(x^2 + 2) D) e^x(x^2 + 4x)

24. What is the fourth derivative of y = cos(2x)?

A) 16 cos(2x) B) 16 sin(2x) C) -16 cos(2x) D) -16 sin(2x)

25. If y = uv, and u = x^3, v = e^x, what is the first term in the expansion of the third derivative using Leibnitz Theorem?

A) x^3 * e^x B) 3x^2 * e^x C) 6x * e^x D) 6 * e^x

26. The nth derivative of y = 1/(1-x) is:

A) n! / (1-x)^(n+1) B) n! / (1-x)^n C) (n+1)! / (1-x)^(n+1) D) 1 / (1-x)^(n+1)

27. What is the value of the second derivative of y = x log(x) at x=e?

A) 1/e B) 0 C) 1 D) 2/e

28. If y = uv, where u = x^2 and v = cos(x), find the third derivative using Leibnitz Theorem.

A) x^2 cos(x) - 6x sin(x) - 6 cos(x) B) x^2 cos(x) + 6x sin(x) - 6 cos(x) C) x^2 cos(x) - 6x sin(x) + 6 cos(x) D) x^2 cos(x) + 6x sin(x) + 6 cos(x)

29. What is the nth derivative of y = x^n when n is a positive integer?

A) n! B) n C) 0 D) 1

30. If y = e^(ax) * cos(bx), what is the first derivative?

A) e^(ax)[(a cos(bx) - b sin(bx))] B) e^(ax)[(a cos(bx) + b sin(bx))] C) e^(ax)[(b cos(bx) - a sin(bx))] D) e^(ax)[(b cos(bx) + a sin(bx))]

31. What is the third derivative of y = x^3 - 2x^2 + 5x - 1?

A) 6x - 4 B) 6 C) 0 D) 1

32. Consider y = uv. If u = x and v = ln(x), find the second derivative using Leibnitz Theorem.

A) x * (1/x) + 2 * (-1/x^2) B) x * (-1/x^2) + 2 * (1/x) C) x * (1/x) + 1 * (-1/x^2) D) x * (-1/x^2) + 1 * (1/x)

33. The second derivative of y = sinh(x) is:

A) cosh(x) B) sinh(x) C) -sinh(x) D) -cosh(x)

34. If y = uv, and u = x^2, v = sin(x), what is the second term (r=1) in the expansion of the third derivative using Leibnitz Theorem?

A) C(3,1) * (2x) * (3 cos(x)) B) C(3,1) * (2x) * (-sin(x)) C) C(3,1) * (2) * (-sin(x)) D) C(3,1) * (x^2) * (-sin(x))

35. What is the nth derivative of y = x^m, where m < n?

A) 0 B) m! C) x^(m-n) D) Undefined

36. If y = x^3 * e^x, find the third derivative using Leibnitz Theorem.

A) e^x(x^3 + 9x^2 + 27x + 6) B) e^x(x^3 + 3x^2 + 3x + 1) C) e^x(x^3 + 9x^2 + 18x + 6) D) e^x(x^3 + 6x^2 + 9x + 3)

37. What is the value of the second derivative of y = x^3 ln(x) at x=1?

A) 1 B) 2 C) 3 D) 0

38. If y = uv, and u = x^n, v = e^x, find the (n+1)th derivative using Leibnitz Theorem.

A) e^x * sum_{r=0}^{n} C(n+1,r) * x^(n-r) B) e^x * sum_{r=0}^{n} C(n+1,r) * (n-r)! * x^(n-r) C) e^x * sum_{r=0}^{n} C(n+1,r) * (n-r) * x^(n-r) D) e^x * sum_{r=0}^{n} C(n+1,r) * r! * x^(n-r)

39. What is the second derivative of y = x^2 sin(x)?

A) x^2 sin(x) + 4x cos(x) - 3 sin(x) B) x^2 sin(x) - 4x cos(x) + 3 sin(x) C) x^2 sin(x) + 4x cos(x) + 3 sin(x) D) x^2 cos(x) + 4x sin(x) - 3 cos(x)

40. If y = log(1+x), what is the third derivative at x=0?

A) 1 B) 2 C) -2 D) 0

41. Leibnitz Theorem can be applied to find the nth derivative of a function that is:

A) A single function B) A sum of functions C) A product of two functions D) An inverse trigonometric function

42. What is the nth derivative of y = sin(ax+b)?

A) a^n sin(ax+b + n*pi/2) B) a^n cos(ax+b + n*pi/2) C) a^n sin(ax+b) D) a^n sin(ax+b + pi/2)

43. If y = uv, and u = x^2, v = e^x, find the third derivative using Leibnitz Theorem.

A) e^x(x^2 + 6x + 6) B) e^x(x^2 + 3x + 1) C) e^x(x^2 + 6x + 3) D) e^x(x^2 + 4x + 2)

44. What is the second derivative of y = x^2 * cos(x)?

A) x^2 cos(x) + 4x sin(x) - 3 cos(x) B) x^2 cos(x) - 4x sin(x) - 3 cos(x) C) x^2 cos(x) - 4x sin(x) + 3 cos(x) D) x^2 sin(x) + 4x cos(x) - 3 sin(x)

45. If y = uv, where u = x and v = x^2, what is the third derivative of y?

A) 6x B) 6 C) 0 D) 1

46. The nth derivative of y = cos(ax+b) is:

A) a^n cos(ax+b + n*pi/2) B) a^n sin(ax+b + n*pi/2) C) a^n cos(ax+b) D) a^n cos(ax+b + pi/2)

47. Let y = x^3. What is the third derivative of y?

A) x^3 B) 3x^2 C) 6x D) 6

48. If y = uv, and u = x^2, v = ln(x), find the third derivative using Leibnitz Theorem.

A) x^2 * (-2/x^3) + 3 * (2x) * (-1/x^2) + 3 * (2) * (1/x) B) x^2 * (1/x) + 3 * (2x) * (-1/x^2) + 3 * (2) * (1/x) C) x^2 * (-1/x^2) + 3 * (2x) * (1/x) + 3 * (2) * (-1/x^2) D) x^2 * (1/x) + 3 * (2x) * (1/x) + 3 * (2) * (-1/x^2)

49. What is the second derivative of y = x^n?

A) n(n-1)x^(n-2) B) n x^(n-1) C) n^2 x^(n-2) D) n(n-2)x^(n-1)

50. If y = uv, and u = x, v = e^(2x), find the second derivative using Leibnitz Theorem.

A) e^(2x) * (4x + 4) B) e^(2x) * (2x + 2) C) e^(2x) * (4x + 2) D) e^(2x) * (2x + 4)

Higher Order Derivatives and Leibnitz Theorem - Online Test

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