Chain Rule: Mastering Calculus for Competitive Exams

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Chain Rule

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Chain Rule MCQ & Objective Questions

The Chain Rule is a fundamental concept in calculus that plays a crucial role in various examinations. Mastering this topic not only enhances your understanding of derivatives but also boosts your confidence in solving complex problems. Practicing MCQs and objective questions on the Chain Rule is essential for effective exam preparation, helping you identify important questions and improve your problem-solving speed.

What You Will Practise Here

Understanding the definition and significance of the Chain Rule
Application of the Chain Rule in differentiation
Identifying functions suitable for the Chain Rule
Solving problems involving multiple functions
Using the Chain Rule in real-world scenarios
Common derivatives involving the Chain Rule
Diagrams illustrating the Chain Rule concept

Exam Relevance

The Chain Rule is frequently tested in various examinations, including CBSE, State Boards, NEET, and JEE. Students can expect to encounter questions that require them to apply the Chain Rule to differentiate composite functions. Common question patterns include direct application problems, as well as those that involve multiple steps and require a clear understanding of the underlying concepts.

Common Mistakes Students Make

Confusing the order of differentiation when applying the Chain Rule
Neglecting to identify all functions involved in a composite function
Overlooking the need to simplify expressions before differentiating
Misapplying the Chain Rule to functions that do not require it

FAQs

Question: What is the Chain Rule in calculus?
Answer: The Chain Rule is a formula used to compute the derivative of a composite function, allowing you to differentiate functions that are nested within each other.

Question: How can I practice Chain Rule MCQ questions effectively?
Answer: You can practice Chain Rule MCQ questions by solving objective questions available on educational platforms like SoulShift, which provide a variety of practice questions tailored for exam preparation.

Don't wait any longer! Start solving Chain Rule practice MCQs today to solidify your understanding and excel in your exams. Your success is just a question away!

Q. If a man can do a piece of work in 15 days, how much work can he do in 3 days?

A. 1/5
B. 1/4
C. 1/3
D. 1/2

Solution

In 1 day, he does 1/15 of the work. In 3 days, he does 3/15 = 1/5 of the work.

Correct Answer: B — 1/4

Q. If a man can finish a work in 25 days, how long will it take for him to finish 40% of the work?

A. 10 days
B. 12 days
C. 15 days
D. 20 days

Solution

40% of the work = 0.4 * 25 days = 10 days.

Correct Answer: B — 12 days

Q. If a man can walk 5 km in 1 hour, how long will it take him to walk 20 km?

A. 3 hours
B. 4 hours
C. 5 hours
D. 6 hours

Solution

Time = Distance / Speed = 20 km / 5 km/h = 4 hours.

Correct Answer: B — 4 hours

Q. If y = (2x + 1)^3, find dy/dx at x = 1.

A. 12
B. 18
C. 24
D. 30

Solution

dy/dx = 6(2x + 1)^2. At x = 1, dy/dx = 6(2(1) + 1)^2 = 6(3)^2 = 54.

Correct Answer: A — 12

Q. If y = (2x + 1)^3, find dy/dx.

A. 6(2x + 1)^2
B. 3(2x + 1)^2
C. 2(2x + 1)^2
D. 12(2x + 1)^2

Solution

dy/dx = 6(2x + 1)^2 by the chain rule.

Correct Answer: A — 6(2x + 1)^2

Q. If y = (x^2 + 1)^5, find dy/dx at x = 1.

A. 10
B. 20
C. 30
D. 40

Solution

dy/dx = 5(x^2 + 1)^4(2x). At x = 1, dy/dx = 5(2)^4(2) = 5(16)(2) = 160.

Correct Answer: B — 20

Q. If y = (x^2 + 1)^5, find dy/dx.

A. 10x(x^2 + 1)^4
B. 5(x^2 + 1)^4
C. 5x(x^2 + 1)^4
D. 20x(x^2 + 1)^3

Solution

dy/dx = 10x(x^2 + 1)^4 by the chain rule.

Correct Answer: A — 10x(x^2 + 1)^4

Q. If y = 2^x, find dy/dx at x = 1.

A. 0.693
B. 1.386
C. 2.718
D. 3.141

Solution

dy/dx = 2^x * ln(2). At x = 1, dy/dx = 2^1 * ln(2) = 2 * 0.693 = 1.386.

Correct Answer: A — 0.693

Q. If y = 3x^2 + 2x, find dy/dx at x = 2.

A. 14
B. 18
C. 22
D. 26

Solution

First, find dy/dx = 6x + 2. At x = 2, dy/dx = 6(2) + 2 = 12 + 2 = 14.

Correct Answer: B — 18

Q. If y = 4x^2 + 3x + 2, find dy/dx at x = -1.

A. -5
B. -3
C. 1
D. 5

Solution

dy/dx = 8x + 3. At x = -1, dy/dx = 8(-1) + 3 = -8 + 3 = -5.

Correct Answer: A — -5

Q. If y = 4x^3 - 2x + 1, find dy/dx at x = -1.

A. -10
B. -8
C. -6
D. -4

Solution

dy/dx = 12x^2 - 2. At x = -1, dy/dx = 12(-1)^2 - 2 = 12 - 2 = 10.

Correct Answer: C — -6

Q. If y = 5x^4 + 3x^2 - x, find dy/dx at x = 1.

A. 20
B. 22
C. 24
D. 26

Solution

dy/dx = 20x^3 + 6x - 1. At x = 1, dy/dx = 20(1)^3 + 6(1) - 1 = 20 + 6 - 1 = 25.

Correct Answer: B — 22

Q. If y = 5x^4 - 3x^3 + 2x - 1, find dy/dx at x = 1.

A. 14
B. 16
C. 18
D. 20

Solution

dy/dx = 20x^3 - 9x^2 + 2. At x = 1, dy/dx = 20 - 9 + 2 = 13.

Correct Answer: B — 16

Q. If y = cos(5x^2), find dy/dx.

A. -10xsin(5x^2)
B. -5xsin(5x^2)
C. -25xsin(5x^2)
D. -2xsin(5x^2)

Solution

dy/dx = -10xsin(5x^2) by the chain rule.

Correct Answer: A — -10xsin(5x^2)

Q. If y = e^(3x), find dy/dx at x = 0.

A. 1
B. 3
C. e
D. 3e

Solution

dy/dx = 3e^(3x). At x = 0, dy/dx = 3e^(0) = 3.

Correct Answer: A — 1

Q. If y = e^(3x), find dy/dx.

A. 3e^(3x)
B. e^(3x)
C. 9e^(3x)
D. 6e^(3x)

Solution

dy/dx = 3e^(3x) by the chain rule.

Correct Answer: A — 3e^(3x)

Q. If y = ln(5x^2 + 3), find dy/dx at x = 1.

A. 5/8
B. 3/8
C. 1/8
D. 1/5

Solution

dy/dx = (10x)/(5x^2 + 3). At x = 1, dy/dx = (10)/(5(1) + 3) = 10/8 = 5/4.

Correct Answer: A — 5/8

Q. If y = ln(5x^2 + 3), find dy/dx.

A. 10/(5x^2 + 3)
B. 5/(5x^2 + 3)
C. 2/(5x^2 + 3)
D. 15/(5x^2 + 3)

Solution

dy/dx = (10x)/(5x^2 + 3) by the chain rule.

Correct Answer: A — 10/(5x^2 + 3)

Q. If y = sin(2x), find dy/dx at x = π/4.

A. 0
B. 1
C. √2/2
D. √2

Solution

dy/dx = 2cos(2x). At x = π/4, dy/dx = 2cos(π/2) = 2(0) = 0.

Correct Answer: D — √2

Q. If y = sqrt(4x^2 + 1), find dy/dx.

A. (4x)/(sqrt(4x^2 + 1))
B. (2x)/(sqrt(4x^2 + 1))
C. (8x)/(sqrt(4x^2 + 1))
D. (2)/(sqrt(4x^2 + 1))

Solution

Using the chain rule, dy/dx = (4x)/(2sqrt(4x^2 + 1)) = (4x)/(sqrt(4x^2 + 1)).

Correct Answer: A — (4x)/(sqrt(4x^2 + 1))

Q. If y = sqrt(x^2 + 1), find dy/dx at x = 0.

A. 0
B. 1
C. 1/2
D. 1/√2

Solution

dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x). At x = 0, dy/dx = (1/2)(1)^(-1/2)(0) = 0.

Correct Answer: B — 1

Q. If y = tan(3x), find dy/dx at x = π/6.

A. 3√3
B. 3
C. √3
D. 1

Solution

dy/dx = 3sec^2(3x). At x = π/6, dy/dx = 3sec^2(π/2) = 3(∞) = undefined.

Correct Answer: A — 3√3

Q. If y = tan(3x), find dy/dx.

A. 3sec^2(3x)
B. 3tan^2(3x)
C. sec^2(3x)
D. 3tan(3x)

Solution

Using the chain rule, dy/dx = 3sec^2(3x).

Correct Answer: A — 3sec^2(3x)

Q. If y = x^3 * e^x, find dy/dx at x = 0.

A. 0
B. 1
C. 3
D. 6

Solution

Using product rule, dy/dx = 3x^2 * e^x + x^3 * e^x. At x = 0, dy/dx = 0 + 0 = 0.

Correct Answer: A — 0

Q. If y = √(4x^2 + 1), find dy/dx.

A. 4x/(√(4x^2 + 1))
B. 2x/(√(4x^2 + 1))
C. 2/(√(4x^2 + 1))
D. 8x/(√(4x^2 + 1))

Solution

dy/dx = (4x)/(2√(4x^2 + 1)) = 4x/(√(4x^2 + 1)).

Correct Answer: A — 4x/(√(4x^2 + 1))

Q. If y = √(x^2 + 1), find dy/dx at x = 1.

A. 1/√2
B. 1/2
C. 1
D. 2

Solution

dy/dx = (1/2)(x^2 + 1)^(-1/2)(2x). At x = 1, dy/dx = (1/2)(2)^(-1/2)(2) = 1/√2.

Correct Answer: A — 1/√2

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