Q. Determine the derivative of f(x) = x^3 - 4x + 7. (2023)
A.
3x^2 - 4
B.
3x^2 + 4
C.
x^2 - 4
D.
3x^2 - 7
Show solution
Solution
Using the power rule, f'(x) = 3x^2 - 4.
Correct Answer:
A
— 3x^2 - 4
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Q. Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)
A.
5x^4 - 9x^2 + 2
B.
5x^4 - 9x + 2
C.
5x^4 - 3x^2 + 2
D.
5x^4 - 3x^3
Show solution
Solution
Using the power rule, f'(x) = 5x^4 - 9x^2 + 2.
Correct Answer:
A
— 5x^4 - 9x^2 + 2
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Q. Differentiate f(x) = 4x^2 * e^x. (2022)
A.
4e^x + 4x^2e^x
B.
4x^2e^x + 4xe^x
C.
4e^x + 2x^2e^x
D.
8xe^x
Show solution
Solution
Using the product rule, f'(x) = 4e^x + 4x^2e^x.
Correct Answer:
A
— 4e^x + 4x^2e^x
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Q. Differentiate f(x) = 4x^2 + 3x - 5. (2019)
A.
8x + 3
B.
4x + 3
C.
2x + 3
D.
8x - 3
Show solution
Solution
Using the power rule, f'(x) = 8x + 3.
Correct Answer:
A
— 8x + 3
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Q. Differentiate f(x) = 4x^5 - 2x^3 + x. (2022)
A.
20x^4 - 6x^2 + 1
B.
20x^4 - 6x^2
C.
4x^4 - 2x^2 + 1
D.
5x^4 - 2x^2
Show solution
Solution
Using the power rule, f'(x) = 20x^4 - 6x^2 + 1.
Correct Answer:
A
— 20x^4 - 6x^2 + 1
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Q. Differentiate f(x) = ln(x^2 + 1). (2022)
A.
2x/(x^2 + 1)
B.
1/(x^2 + 1)
C.
2x/(x^2 - 1)
D.
x/(x^2 + 1)
Show solution
Solution
Using the chain rule, f'(x) = 2x/(x^2 + 1).
Correct Answer:
A
— 2x/(x^2 + 1)
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Q. Differentiate f(x) = x^2 * e^x. (2022)
A.
x^2 * e^x + 2x * e^x
B.
2x * e^x + x^2 * e^x
C.
x^2 * e^x + e^x
D.
2x * e^x
Show solution
Solution
Using the product rule, f'(x) = x^2 * e^x + 2x * e^x.
Correct Answer:
A
— x^2 * e^x + 2x * e^x
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Q. Differentiate f(x) = x^2 * ln(x).
A.
2x * ln(x) + x
B.
x * ln(x) + 2x
C.
2x * ln(x)
D.
x^2/x
Show solution
Solution
Using the product rule, f'(x) = 2x * ln(x) + x.
Correct Answer:
A
— 2x * ln(x) + x
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Q. Differentiate the function f(x) = ln(x^2 + 1).
A.
2x/(x^2 + 1)
B.
2/(x^2 + 1)
C.
1/(x^2 + 1)
D.
x/(x^2 + 1)
Show solution
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. Differentiate the function f(x) = x^2 * e^x.
A.
x^2 * e^x + 2x * e^x
B.
2x * e^x + x^2 * e^x
C.
x^2 * e^x + e^x
D.
2x * e^x + e^x
Show solution
Solution
Using the product rule, f'(x) = (x^2)' * e^x + x^2 * (e^x)' = 2x * e^x + x^2 * e^x.
Correct Answer:
A
— x^2 * e^x + 2x * e^x
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Q. Find the derivative of f(x) = 4x^3 - 2x + 1. (2022)
A.
12x^2 - 2
B.
12x^2 + 2
C.
4x^2 - 2
D.
4x^2 + 2
Show solution
Solution
Using the power rule, f'(x) = 12x^2 - 2.
Correct Answer:
A
— 12x^2 - 2
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Q. Find the derivative of f(x) = 5x^2 + 3x - 1. (2020)
A.
10x + 3
B.
5x + 3
C.
10x - 1
D.
5x^2 + 3
Show solution
Solution
Using the power rule, f'(x) = 10x + 3.
Correct Answer:
A
— 10x + 3
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Q. Find the derivative of f(x) = 5x^2 + 3x - 7. (2020)
A.
10x + 3
B.
5x + 3
C.
10x - 3
D.
5x - 3
Show solution
Solution
Using the power rule, f'(x) = 10x + 3.
Correct Answer:
A
— 10x + 3
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Q. Find the derivative of f(x) = 5x^3 - 4x + 7. (2019)
A.
15x^2 - 4
B.
15x^2 + 4
C.
5x^2 - 4
D.
5x^2 + 4
Show solution
Solution
Using the power rule, f'(x) = 15x^2 - 4.
Correct Answer:
A
— 15x^2 - 4
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Q. Find the derivative of f(x) = x^3 * ln(x). (2023)
A.
3x^2 * ln(x) + x^2
B.
3x^2 * ln(x) + x^3/x
C.
3x^2 * ln(x) + x^3
D.
3x^2 * ln(x) + 1
Show solution
Solution
Using the product rule, f'(x) = (x^3)' * ln(x) + x^3 * (ln(x))' = 3x^2 * ln(x) + x^2.
Correct Answer:
A
— 3x^2 * ln(x) + x^2
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Q. Find the derivative of f(x) = x^4 + 2x^3 - x + 1. (2023)
A.
4x^3 + 6x^2 - 1
B.
4x^3 + 2x^2 - 1
C.
3x^3 + 6x^2 - 1
D.
4x^3 + 2x - 1
Show solution
Solution
Using the power rule, f'(x) = 4x^3 + 6x^2 - 1.
Correct Answer:
A
— 4x^3 + 6x^2 - 1
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Q. Find the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 2.
A.
4x^3 - 12x^2 + 12x
B.
4x^3 - 12x + 6
C.
12x^2 - 4x + 6
D.
4x^3 - 12x^2 + 2
Show solution
Solution
Using the power rule, f'(x) = 4x^3 - 12x^2 + 12x.
Correct Answer:
A
— 4x^3 - 12x^2 + 12x
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Q. Find the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 24x + 5. (2023)
A.
4x^3 - 12x^2 + 12x - 24
B.
4x^3 - 12x^2 + 6x - 24
C.
4x^3 - 12x^2 + 12x
D.
4x^3 - 12x^2 + 6x
Show solution
Solution
Using the power rule, f'(x) = 4x^3 - 12x^2 + 12x - 24.
Correct Answer:
A
— 4x^3 - 12x^2 + 12x - 24
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Q. Find the derivative of f(x) = x^5 - 2x^3 + x. (2019)
A.
5x^4 - 6x^2 + 1
B.
5x^4 - 6x
C.
5x^4 + 2x^2 + 1
D.
5x^4 - 2x^2
Show solution
Solution
Using the power rule, f'(x) = 5x^4 - 6x^2 + 1.
Correct Answer:
A
— 5x^4 - 6x^2 + 1
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Q. Find the derivative of g(x) = sin(x) + cos(x). (2020)
A.
cos(x) - sin(x)
B.
-sin(x) - cos(x)
C.
sin(x) + cos(x)
D.
-cos(x) + sin(x)
Show solution
Solution
Using the derivatives of sine and cosine, g'(x) = cos(x) - sin(x).
Correct Answer:
A
— cos(x) - sin(x)
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Q. If f(x) = 3x^2 + 2x, what is f'(2)? (2023)
Show solution
Solution
First, find f'(x) = 6x + 2. Then, f'(2) = 6(2) + 2 = 12 + 2 = 14.
Correct Answer:
A
— 10
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Q. If f(x) = 4x^3 - 2x^2 + x, what is f''(x)?
A.
24x - 4
B.
12x - 2
C.
12x - 4
D.
24x - 2
Show solution
Solution
First, find f'(x) = 12x^2 - 4x + 1, then differentiate again to get f''(x) = 24x - 4.
Correct Answer:
A
— 24x - 4
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Q. If f(x) = 5x^2 + 3x - 1, what is f''(x)? (2020)
Show solution
Solution
The first derivative f'(x) = 10x + 3, and the second derivative f''(x) = 10.
Correct Answer:
A
— 10
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Q. If f(x) = 5x^2 + 3x - 1, what is f'(2)? (2020)
Show solution
Solution
First, find f'(x) = 10x + 3. Then, f'(2) = 10(2) + 3 = 20 + 3 = 23.
Correct Answer:
A
— 27
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Q. If f(x) = 5x^2 - 3x + 7, what is f''(x)? (2020)
Show solution
Solution
The first derivative f'(x) = 10x - 3, and the second derivative f''(x) = 10.
Correct Answer:
A
— 10
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Q. If f(x) = x^2 * e^x, find f'(x). (2019)
A.
e^x(x^2 + 2x)
B.
e^x(x^2 - 2x)
C.
x^2 * e^x
D.
2x * e^x
Show solution
Solution
Using the product rule, f'(x) = e^x(x^2 + 2x).
Correct Answer:
A
— e^x(x^2 + 2x)
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Q. If f(x) = x^2 * e^x, what is f'(x)? (2019)
A.
e^x(x^2 + 2x)
B.
e^x(x^2 - 2x)
C.
2xe^x
D.
x^2e^x
Show solution
Solution
Using the product rule, f'(x) = e^x(x^2 + 2x).
Correct Answer:
A
— e^x(x^2 + 2x)
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Q. If f(x) = x^2 * ln(x), what is f'(x)? (2022)
A.
2x * ln(x) + x
B.
x * ln(x) + 2x
C.
2x * ln(x) - x
D.
x * ln(x) - 2x
Show solution
Solution
Using the product rule, f'(x) = 2x * ln(x) + x.
Correct Answer:
A
— 2x * ln(x) + x
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Q. If f(x) = x^2 + 3x + 5, what is f''(x)? (2020)
Show solution
Solution
The first derivative f'(x) = 2x + 3, and the second derivative f''(x) = 2.
Correct Answer:
A
— 2
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Q. If f(x) = x^3 - 4x + 1, what is f''(x)? (2023)
A.
6x - 4
B.
6x + 4
C.
3x^2 - 4
D.
3x^2 + 4
Show solution
Solution
First derivative f'(x) = 3x^2 - 4, then f''(x) = 6x.
Correct Answer:
A
— 6x - 4
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Showing 1 to 30 of 43 (2 Pages)
Differentiation Rules MCQ & Objective Questions
Differentiation Rules are a crucial part of calculus that every student must master for success in exams. Understanding these rules not only helps in solving complex problems but also enhances your ability to tackle objective questions effectively. Practicing MCQs and other practice questions on Differentiation Rules can significantly improve your exam preparation and boost your confidence in handling important questions.
What You Will Practise Here
Basic concepts of differentiation and its significance
Product and quotient rules for differentiation
Chain rule and its applications in various problems
Higher-order derivatives and their relevance
Implicit differentiation techniques
Applications of differentiation in real-life scenarios
Common derivatives of standard functions
Exam Relevance
The topic of Differentiation Rules is frequently tested in CBSE, State Boards, NEET, and JEE examinations. Students can expect a variety of question patterns, including direct application of rules, conceptual understanding, and problem-solving scenarios. Mastery of this topic is essential, as it forms the foundation for many advanced concepts in calculus and is often linked to scoring well in competitive exams.
Common Mistakes Students Make
Confusing the product rule with the quotient rule
Neglecting to apply the chain rule correctly in composite functions
Overlooking the importance of simplifying expressions before differentiation
Misinterpreting implicit differentiation scenarios
Failing to memorize common derivatives, leading to errors in calculations
FAQs
Question: What are the basic rules of differentiation?Answer: The basic rules include the power rule, product rule, quotient rule, and chain rule, which help in finding derivatives of various functions.
Question: How can I improve my understanding of differentiation?Answer: Regular practice of MCQs and solving important Differentiation Rules questions for exams will enhance your understanding and retention of concepts.
Now is the time to take charge of your learning! Dive into our practice MCQs on Differentiation Rules and test your understanding. Remember, consistent practice is the key to mastering this essential topic and achieving your academic goals.
