Higher Order Derivatives and Leibnitz Theorem - Question Bank

1. What is the value of the fourth derivative of y = x^5 at x=1?
A) 120
B) 24
C) 60
D) 5
2. 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)
3. 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)
4. 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)
5. Let y = x^3. What is the third derivative of y?
A) x^3
B) 3x^2
C) 6x
D) 6
6. 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)
7. 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
8. 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)
9. 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)
10. 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)
11. 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
12. If y = log(1+x), what is the third derivative at x=0?
A) 1
B) 2
C) -2
D) 0
13. 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)
14. 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)
15. 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
16. 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)
17. What is the nth derivative of y = x^m, where m < n?
A) 0
B) m!
C) x^(m-n)
D) Undefined
18. 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))
19. The second derivative of y = sinh(x) is:
A) cosh(x)
B) sinh(x)
C) -sinh(x)
D) -cosh(x)
20. 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)
21. What is the third derivative of y = x^3 - 2x^2 + 5x - 1?
A) 6x - 4
B) 6
C) 0
D) 1
22. 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))]
23. What is the nth derivative of y = x^n when n is a positive integer?
A) n!
B) n
C) 0
D) 1
24. 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)
25. 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
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. 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
28. 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)
29. 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)
30. Which of the following functions has a third derivative that is identically zero?
A) x^4
B) x^3
C) x^2
D) x
31. If y = tan(x), what is the second derivative at x=0?
A) 0
B) 1
C) 2
D) tan(0)
32. What is the nth derivative of y = x^n?
A) n!
B) n
C) 1
D) 0
33. 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)
34. What is the value of the third derivative of y = x^4 at x=2?
A) 24
B) 48
C) 96
D) 192
35. Leibnitz Theorem is a generalization of which rule?
A) Chain Rule
B) Product Rule
C) Quotient Rule
D) Sum Rule
36. 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
37. What is the nth derivative of a constant function?
A) The constant itself
B) 0
C) 1
D) Undefined
38. 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)
39. 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
40. What is the third derivative of y = cos(x)?
A) sin(x)
B) cos(x)
C) -sin(x)
D) -cos(x)
41. 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
42. 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)
43. What is the value of the second derivative of y = ln(x) at x=1?
A) 1
B) -1
C) 0
D) 2
44. If y = x^3, what is the fourth derivative of y?
A) 6x
B) 6
C) 0
D) 1
45. 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
46. 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)
47. 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
48. 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)
49. What is the third derivative of y = sin(x)?
A) cos(x)
B) -sin(x)
C) -cos(x)
D) sin(x)
50. 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
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