Differentiability and derivatives up to second order - One Line Questions
1.
If f(x) = 1/x, then f''(x) is: —
2/x^3
2.
If f(x) = sin(x), then f'(x) is: —
cos(x)
3.
What is the derivative of f(x) = cot(x)? —
-csc^2(x)
4.
If f(x) = e^(-x), then f''(x) is: —
e^(-x)
5.
Let f(x) = cos(x^2). What is f''(x)? —
-2*sin(x^2) - 4x^2*cos(x^2)
6.
If f(x) = cos(x), then f''(x) is: —
-cos(x)
7.
If f(x) = x^(1/2), then f'(x) is: —
(1/2)x^(-1/2)
8.
Chain rule for differentiation states that if y = f(u) and u = g(x), then dy/dx is: —
(dy/du) * (du/dx)
9.
If f(x) = |x|, then f'(0) is: —
Does not exist
10.
If f(x) = |x-2|, f'(2) is: —
Does not exist
11.
What is the derivative of f(x) = arctan(x)? —
1 / (1 + x^2)
12.
If f(x) = arctan(x^2), then f'(x) is: —
2x / (1 + x^4)
13.
If f(x) = sqrt(x), then f'(x) is: —
1 / (2*sqrt(x))
14.
The derivative of f(x) = arcsin(x) is: —
1 / sqrt(1 - x^2)
15.
If f(x) = sin^(-1)(x), then f'(x) is: —
1 / sqrt(1 - x^2)
16.
If f(x) = ln(2x), then f'(x) is: —
1/x
17.
The derivative of f(x) = log_a(x) is: —
1/(x*ln(a))
18.
If f(x) = ln(x^2), then f'(x) is: —
2/x
19.
If f(x) = x^2 * sin(x), then f'(x) is: —
2x * sin(x) + x^2 * cos(x)
20.
If f(x) = x^2 + 3x + 2, then f''(x) is: —
2
21.
What is the derivative of f(x) = x^2 * cos(x)? —
2x*cos(x) - x^2*sin(x)
22.
What is the second derivative of f(x) = x^3? —
6x
23.
Let f(x) = x^3. What is f'''(x)? —
6
24.
Let f(x) = x^3 - 2x^2 + 5x - 1. What is f''(x)? —
6x - 4
25.
What is the derivative of f(x) = x^3 / sin(x)? —
(3x^2*sin(x) - x^3*cos(x)) / sin^2(x)
26.
Let f(x) = sin(x^3). What is f''(x)? —
6x*cos(x^3) - 9x^4*sin(x^3)
27.
What is the second derivative of f(x) = x^4? —
12x^2
28.
The second derivative of f(x) = 5 is: —
0
29.
If f(x) = x^5, then f'''(x) is: —
60x^2
30.
The derivative of f(x) = a^x is: —
a^x * ln(a)
31.
If f(x) = x^a, then f'(x) is: —
ax^(a-1)
32.
If f(x) = c * g(x), where c is a constant, then f'(x) is: —
c * g'(x)
33.
If f(x) = sin(2x), then f'(x) is: —
2*cos(2x)
34.
Let f(x) = e^(3x). What is f'(x)? —
3 * e^(3x)
35.
The derivative of f(x) = e^x is: —
e^x
36.
If f(x) = x * e^x, then f'(x) is: —
e^x + x*e^x
37.
The function f(x) = x^2 is differentiable everywhere because: —
It is a polynomial.
38.
A function is not differentiable at a point if: —
All of the above.
39.
What is the second derivative of f(x) = e^(kx)? —
k^2*e^(kx)
40.
A function f(x) is differentiable at x=c if and only if: —
lim (h->0) [f(c+h) - f(c)] / h exists and is finite.
41.
If f(x) = x*ln(x), then f'(x) is: —
1 + ln(x)
42.
If f(x) = x^n, then f''(x) is: —
n*(n-1)*x^(n-2)
43.
The derivative of f(x) = x^n is: —
nx^(n-1)
44.
What is the derivative of f(x) = tan(x)? —
sec^2(x)
45.
If f(x) = tan^(-1)(x), then f'(x) is: —
1 / (1 + x^2)
46.
What is the derivative of f(x) = sec(x)? —
sec(x)tan(x)
47.
The derivative of a constant function is: —
0
48.
The function f(x) = |x| + |x-1| is differentiable at x=0 if: —
The function is defined at x=0.
49.
A function f(x) is differentiable at a point x=c if: —
The limit [f(c+h) - f(c)] / h as h approaches 0 exists.
50.
The product rule for differentiation states that if f(x) = u(x) * v(x), then f'(x) =: —
u(x) * v'(x) + v(x) * u'(x)