Higher Order Derivatives and Leibnitz Theorem - One Line Questions
1.
If y = log_a(x), what is the second derivative of y? —
-1/(x^2 * ln a)
2.
If y = a^x, what is the nth derivative of y? —
(ln a)^n * a^x
3.
If y = tan(x), what is the second derivative at x=0? —
2
4.
What is the nth derivative of y = x^m, where m < n? —
0
5.
What is the value of the second derivative of y = ln(x) at x=1? —
-1
6.
What is the value of the second derivative of y = x^3 ln(x) at x=1? —
3
7.
If y = log(1+x), what is the third derivative at x=0? —
2
8.
What is the value of the second derivative of y = x log(x) at x=e? —
1
9.
What is the value of the fourth derivative of y = x^5 at x=1? —
120
10.
What is the fourth derivative of y = cos(2x)? —
16 cos(2x)
11.
What is the value of the third derivative of y = x^4 at x=2? —
96
12.
If y = x^3, what is the fourth derivative of y? —
0
13.
If y = uv, where u = x and v = x^2, what is the third derivative of y? —
6
14.
What is the third derivative of y = x^3 - 2x^2 + 5x - 1? —
6
15.
Leibnitz Theorem can be applied to find the nth derivative of a function that is: —
A product of two functions
16.
The nth derivative of y = cos(ax+b) is: —
a^n cos(ax+b + n*pi/2)
17.
What is the nth derivative of y = sin(ax+b)? —
a^n sin(ax+b + n*pi/2)
18.
If y = ax^n, what is the second derivative, d^2y/dx^2? —
a n(n-1)x^(n-2)
19.
Consider the function y = x^m. For what value of m is the (m+1)th derivative equal to zero? —
Any positive integer m
20.
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? —
C(3,1) * (2x) * (-sin(x))
21.
Leibnitz Theorem is a generalization of which rule? —
Product Rule
22.
What is the third derivative of y = sin(x)? —
-cos(x)
23.
The second derivative of y = sinh(x) is: —
sinh(x)
24.
State the Leibnitz Theorem for the nth derivative of the product of two functions u(x) and v(x). —
d^n(uv)/dx^n = sum_{r=0}^{n} C(n,r) * (d^r u / dx^r) * (d^(n-r) v / dx^(n-r))
25.
If y = uv, and u = x, v = e^(2x), find the second derivative using Leibnitz Theorem. —
e^(2x) * (4x + 4)
26.
If y = e^(ax) * cos(bx), what is the first derivative? —
e^(ax)[(a cos(bx) + b sin(bx))]
27.
If y = uv, and u = x^n, v = e^x, find the (n+1)th derivative using Leibnitz Theorem. —
e^x * sum_{r=0}^{n} C(n+1,r) * x^(n-r)
28.
Let u = x^2 and v = e^x. Using Leibnitz Theorem, find the second derivative of y = uv. —
e^x(x^2 + 4x + 2)
29.
Let y = x^2 * e^x. Find the second derivative using Leibnitz Theorem. —
e^x(x^2 + 4x + 2)
30.
If y = uv, and u = x^2, v = e^x, find the third derivative using Leibnitz Theorem. —
e^x(x^2 + 6x + 6)
31.
If y = x^3 * e^x, find the third derivative using Leibnitz Theorem. —
e^x(x^3 + 9x^2 + 27x + 6)
32.
If f(x) = e^(kx), what is the nth derivative of f(x)? —
k^n * e^(kx)
33.
What is the nth derivative of y = x^n? —
n!
34.
What is the nth derivative of y = x^n when n is a positive integer? —
n!
35.
The nth derivative of y = 1/(1-x) is: —
n! / (1-x)^(n+1)
36.
What is the second derivative of y = x^n? —
n(n-1)x^(n-2)
37.
What is the third derivative of y = cos(x)? —
-sin(x)
38.
What is the nth derivative of a constant function? —
0
39.
Leibnitz Theorem is used to find: —
The nth derivative of a product of two functions
40.
What is the second derivative of a function f(x)? —
The derivative of the first derivative of f(x)
41.
In the Leibnitz Theorem formula, C(n,r) represents: —
The binomial coefficient 'n choose r'
42.
Consider y = uv. If u = x and v = ln(x), find the second derivative using Leibnitz Theorem. —
x * (1/x) + 1 * (-1/x^2)
43.
If y = uv, and u = x, v = sin(x), find the third derivative of y using Leibnitz Theorem. —
x cos(x) - 3 sin(x)
44.
If y = uv, and u = x^2, v = ln(x), find the third derivative using Leibnitz Theorem. —
x^2 * (1/x) + 3 * (2x) * (-1/x^2) + 3 * (2) * (1/x)
45.
If y = x^2 * sin(x), what is the first term in the expansion of the second derivative using Leibnitz Theorem? —
x^2 * sin(x)
46.
If y = uv, where u = x^2 and v = cos(x), find the third derivative using Leibnitz Theorem. —
x^2 cos(x) - 6x sin(x) - 6 cos(x)
47.
What is the second derivative of y = x^2 * cos(x)? —
x^2 cos(x) + 4x sin(x) - 3 cos(x)
48.
What is the second derivative of y = x^2 sin(x)? —
x^2 sin(x) + 4x cos(x) - 3 sin(x)
49.
Let y = x^3. What is the third derivative of y? —
6
50.
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? —
x^3 * e^x