Derivatives (Basics)
Q. Calculate the derivative of f(x) = e^(2x).
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A.
2e^(2x)
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B.
e^(2x)
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C.
2xe^(2x)
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D.
e^(x)
Solution
Using the chain rule, f'(x) = d/dx(e^(2x)) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. Calculate the derivative of f(x) = x^2 * e^x.
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A.
(2x + x^2)e^x
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B.
2xe^x
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C.
x^2e^x
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D.
(x^2 + 2x)e^x
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = (x^2 + 2x)e^x.
Correct Answer: D — (x^2 + 2x)e^x
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Q. Determine the derivative of f(x) = 1/x.
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A.
-1/x^2
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B.
1/x^2
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C.
1/x
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D.
-1/x
Solution
Using the power rule, f'(x) = -1/x^2.
Correct Answer: A — -1/x^2
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Q. Determine the derivative of f(x) = ln(x^2 + 1).
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A.
2x/(x^2 + 1)
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B.
1/(x^2 + 1)
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C.
2/(x^2 + 1)
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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. Determine the derivative of f(x) = x^2 * e^x.
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A.
e^x * (x^2 + 2x)
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B.
e^x * (2x + 1)
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C.
2x * e^x
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D.
x^2 * e^x
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x * (x^2 + 2x).
Correct Answer: A — e^x * (x^2 + 2x)
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Q. Find the derivative of f(x) = 1/x.
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A.
-1/x^2
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B.
1/x^2
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C.
-2/x^2
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D.
1/x
Solution
Using the power rule, f'(x) = -1/x^2.
Correct Answer: A — -1/x^2
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Q. Find the derivative of f(x) = 3x^2 + 5x - 7.
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A.
6x + 5
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B.
3x + 5
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C.
6x - 5
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D.
3x^2 + 5
Solution
Using the power rule, f'(x) = d/dx(3x^2) + d/dx(5x) - d/dx(7) = 6x + 5.
Correct Answer: A — 6x + 5
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Q. Find the derivative of f(x) = 5x^4 - 3x + 2.
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A.
20x^3 - 3
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B.
15x^3 - 3
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C.
20x^4 - 3
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D.
5x^3 - 3
Solution
Using the power rule, f'(x) = 20x^3 - 3.
Correct Answer: A — 20x^3 - 3
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Q. Find the derivative of f(x) = e^(2x).
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A.
2e^(2x)
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B.
e^(2x)
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C.
2xe^(2x)
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D.
e^(x)
Solution
Using the chain rule, f'(x) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. Find the derivative of f(x) = ln(x^2 + 1).
-
A.
2x/(x^2 + 1)
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B.
1/(x^2 + 1)
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C.
2/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. Find the derivative of f(x) = sin(x) at x = π/2.
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A.
0
-
B.
1
-
C.
-1
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D.
undefined
Solution
f'(x) = cos(x); f'(π/2) = cos(π/2) = 0.
Correct Answer: B — 1
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Q. Find the derivative of f(x) = tan(x) at x = 0.
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A.
0
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B.
1
-
C.
undefined
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D.
1/2
Solution
f'(x) = sec^2(x); f'(0) = sec^2(0) = 1.
Correct Answer: B — 1
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Q. Find the derivative of f(x) = tan(x).
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A.
sec^2(x)
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B.
csc^2(x)
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C.
sin^2(x)
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D.
cos^2(x)
Solution
The derivative f'(x) = d/dx(tan(x)) = sec^2(x).
Correct Answer: A — sec^2(x)
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Q. Find 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 - 4x + 6
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C.
3x^2 - 8x
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D.
x^2 - 4x + 6
Solution
Using the power rule, f'(x) = 3x^2 - 8x + 6.
Correct Answer: A — 3x^2 - 8x + 6
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Q. If f(x) = 5x^2 + 3x, what is f'(1)?
Solution
f'(x) = 10x + 3; f'(1) = 10*1 + 3 = 13.
Correct Answer: B — 10
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Q. If f(x) = ln(x), what is f'(x)?
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A.
1/x
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B.
x
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C.
ln(x)
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D.
0
Q. What is the derivative of f(x) = 3x^3 - 5x + 2?
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A.
9x^2 - 5
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B.
3x^2 - 5
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C.
9x^2 + 5
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D.
3x^2 + 5
Solution
f'(x) = 9x^2 - 5.
Correct Answer: A — 9x^2 - 5
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Q. What is the derivative of f(x) = 5x^4 - 3x + 2?
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A.
20x^3 - 3
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B.
20x^3 + 3
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C.
15x^3 - 3
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D.
5x^3 - 3
Solution
The derivative f'(x) = d/dx(5x^4 - 3x + 2) = 20x^3 - 3.
Correct Answer: A — 20x^3 - 3
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Q. What is the derivative of f(x) = e^x?
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A.
e^x
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B.
x*e^x
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C.
1
-
D.
0
Solution
The derivative of e^x is e^x.
Correct Answer: A — e^x
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Q. What is the derivative of f(x) = ln(x^2 + 1)?
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A.
2x/(x^2 + 1)
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B.
1/(x^2 + 1)
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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)?
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A.
cos(x) - sin(x)
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B.
-sin(x) - cos(x)
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C.
sin(x) + cos(x)
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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) = tan(x)?
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A.
sec^2(x)
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B.
csc^2(x)
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C.
sec(x)
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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?
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A.
e^x(2x + x^2)
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B.
e^x(2x)
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C.
e^x(x^2 + 2)
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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
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C.
x + 2
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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 - 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?
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A.
4x^3
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B.
3x^4
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C.
2x^4
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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
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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
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C.
x^4
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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?
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A.
1/(2√x)
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B.
2√x
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C.
1/x
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D.
√x/2
Solution
f'(x) = 1/(2√x).
Correct Answer: A — 1/(2√x)
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