Q. What is the derivative of f(x) = ln(x^2 + 1)?
-
A.
2x/(x^2 + 1)
-
B.
1/(x^2 + 1)
-
C.
2/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. What is the derivative of f(x) = sin(x) + cos(x)?
-
A.
cos(x) - sin(x)
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B.
-sin(x) - cos(x)
-
C.
sin(x) + cos(x)
-
D.
-sin(x) + cos(x)
Solution
Using the derivatives of sine and cosine, f'(x) = cos(x) - sin(x).
Correct Answer: A — cos(x) - sin(x)
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Q. What is the derivative of f(x) = sin(x^2)?
-
A.
2x cos(x^2)
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B.
cos(x^2)
-
C.
2x sin(x^2)
-
D.
sin(x^2)
Solution
Using the chain rule, f'(x) = cos(x^2) * 2x = 2x cos(x^2).
Correct Answer: A — 2x cos(x^2)
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Q. What is the derivative of f(x) = tan(x)?
-
A.
sec^2(x)
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B.
csc^2(x)
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C.
sec(x)
-
D.
tan^2(x)
Solution
The derivative of tan(x) is sec^2(x).
Correct Answer: A — sec^2(x)
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Q. What is the derivative of f(x) = x^2 * e^x?
-
A.
e^x(2x + x^2)
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B.
e^x(2x)
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C.
e^x(x^2 + 2)
-
D.
e^x(x^2 + 1)
Solution
Using the product rule, f'(x) = e^x * (x^2 + 2x).
Correct Answer: A — e^x(2x + x^2)
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Q. What is the derivative of f(x) = x^2 + 2x + 1?
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A.
2x + 2
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B.
2x + 1
-
C.
x + 2
-
D.
2x
Solution
f'(x) = 2x + 2.
Correct Answer: A — 2x + 2
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Q. What is the derivative of f(x) = x^2 at x = 3?
Solution
f'(x) = 2x; f'(3) = 2*3 = 6.
Correct Answer: B — 6
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Q. What is the derivative of f(x) = x^3 - 3x^2 + 4?
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A.
3x^2 - 6x
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B.
3x^2 + 6x
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C.
2x^2 - 6x
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D.
3x^2 - 3
Solution
The derivative f'(x) = 3x^2 - 6x.
Correct Answer: A — 3x^2 - 6x
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Q. What is the derivative of f(x) = x^3 - 3x^2 + 4x - 5?
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A.
3x^2 - 6x + 4
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B.
3x^2 - 6x
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C.
2x^2 - 3x + 4
-
D.
3x^2 + 4
Solution
Using the power rule, f'(x) = 3x^2 - 6x + 4.
Correct Answer: A — 3x^2 - 6x + 4
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Q. What is the derivative of f(x) = x^3 - 3x^2 + 4x?
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A.
3x^2 - 6x + 4
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B.
3x^2 - 6x
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C.
2x^2 - 3x + 4
-
D.
x^2 - 3x + 4
Solution
Using the power rule, f'(x) = 3x^2 - 6x + 4.
Correct Answer: A — 3x^2 - 6x + 4
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Q. What is the derivative of f(x) = x^3 - 4x^2 + 6x?
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A.
3x^2 - 8x + 6
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B.
3x^2 + 8x + 6
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C.
2x^2 - 4x + 6
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D.
3x^2 - 4x + 6
Solution
The derivative f'(x) = d/dx(x^3 - 4x^2 + 6x) = 3x^2 - 8x + 6.
Correct Answer: A — 3x^2 - 8x + 6
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Q. What is the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 24x + 5?
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A.
4x^3 - 12x^2 + 12x - 24
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B.
3x^2 - 12x + 6
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C.
4x^3 - 12x^2 + 6
-
D.
12x^2 - 12
Solution
The derivative f'(x) = 4x^3 - 12x^2 + 12x - 24.
Correct Answer: A — 4x^3 - 12x^2 + 12x - 24
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Q. What is the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1?
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A.
4x^3 - 12x^2 + 12x - 4
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B.
3x^2 - 12x + 6
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C.
4x^3 - 12x^2 + 6
-
D.
12x^2 - 12
Solution
The derivative is f'(x) = 4x^3 - 12x^2 + 12x - 4.
Correct Answer: A — 4x^3 - 12x^2 + 12x - 4
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Q. What is the derivative of f(x) = x^4?
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A.
4x^3
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B.
3x^4
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C.
2x^4
-
D.
x^3
Solution
f'(x) = 4x^3.
Correct Answer: A — 4x^3
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Q. What is the derivative of f(x) = x^5 + 2x^3 - x?
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A.
5x^4 + 6x^2 - 1
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B.
5x^4 + 6x^3 - 1
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C.
5x^4 + 2x^2 - 1
-
D.
5x^4 + 2x^3
Solution
The derivative f'(x) = d/dx(x^5 + 2x^3 - x) = 5x^4 + 6x^2 - 1.
Correct Answer: A — 5x^4 + 6x^2 - 1
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Q. What is the derivative of f(x) = x^5?
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A.
5x^4
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B.
4x^5
-
C.
x^4
-
D.
5x^3
Solution
The derivative f'(x) = d/dx(x^5) = 5x^4.
Correct Answer: A — 5x^4
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Q. What is the derivative of f(x) = |x| at x = 0?
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A.
0
-
B.
1
-
C.
-1
-
D.
Undefined
Solution
The left-hand derivative is -1 and the right-hand derivative is 1, hence the derivative at x = 0 is undefined.
Correct Answer: D — Undefined
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Q. What is the derivative of f(x) = √x?
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A.
1/(2√x)
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B.
2√x
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C.
1/x
-
D.
√x/2
Solution
f'(x) = 1/(2√x).
Correct Answer: A — 1/(2√x)
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Q. What is the derivative of sin(x)?
-
A.
cos(x)
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B.
-cos(x)
-
C.
sin(x)
-
D.
-sin(x)
Solution
The derivative of sin(x) is cos(x).
Correct Answer: A — cos(x)
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Q. What is the derivative of sin^(-1)(x)?
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A.
1/√(1-x^2)
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B.
-1/√(1-x^2)
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C.
1/x
-
D.
0
Solution
The derivative of sin^(-1)(x) is 1/√(1-x^2)
Correct Answer: A — 1/√(1-x^2)
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Q. What is the derivative of x^3?
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A.
3x^2
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B.
2x^3
-
C.
x^2
-
D.
3x
Solution
The derivative of x^3 is 3x^2.
Correct Answer: A — 3x^2
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Q. What is the derivative of y = sin^(-1)(x)?
-
A.
1/√(1-x^2)
-
B.
1/(1+x^2)
-
C.
1/(1-x^2)
-
D.
√(1-x^2)
Solution
The derivative of y = sin^(-1)(x) is 1/√(1-x^2)
Correct Answer: A — 1/√(1-x^2)
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Q. What is the derivative of \( y = \tan^{-1}(x) \)?
-
A.
\( \frac{1}{1+x^2} \)
-
B.
\( \frac{1}{x^2+1} \)
-
C.
\( \frac{1}{x} \)
-
D.
0
Solution
The derivative of \( y = \tan^{-1}(x) \) is \( \frac{1}{1+x^2} \).
Correct Answer: A — \( \frac{1}{1+x^2} \)
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Q. What is the determinant of the matrix [[0, 1], [1, 0]]?
Solution
The determinant is calculated as (0*0) - (1*1) = 0 - 1 = -1.
Correct Answer: C — -1
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Q. What is the determinant of the matrix \( E = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \)?
Solution
The determinant is calculated as \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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Q. What is the determinant of the matrix \( H = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \)?
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)?
Solution
The determinant of the identity matrix is 1.
Correct Answer: B — 1
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \)?
Solution
The determinant is calculated as (1*1) - (1*1) = 1 - 1 = 0.
Correct Answer: A — 0
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \)?
Solution
The determinant is calculated as \( 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1 \).
Correct Answer: B — 1
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \)?
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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