Master Successive Differentiation with Step-by-Step Solutions and Examples

Master successive differentiation with expert solutions. Explore problems and step-by-step answers to improve your differentiation skills. Learn with real examples now!

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Successive differentiation problems involve taking multiple derivatives of a function. It requires precision and practice. This guide provides expert solutions to help you solve these problems effectively. Our step-by-step examples make complex concepts easy to understand and apply, enhancing your calculus skills.

Sample Questions and Answers:

Q: Find the second derivative of f(x) = x^3 + 5x^2 - 3x + 1.
A: The first derivative is f'(x) = 3x^2 + 10x - 3, and the second derivative is f''(x) = 6x + 10.

Q: Differentiate the function f(x) = sin(x) + cos(x).
A: The first derivative is f'(x) = cos(x) - sin(x).

Q: Find the third derivative of f(x) = e^x(x^2 + 3x).
A: The first derivative is f'(x) = e^x(2x + 3) + e^x(x^2 + 3x), and the third derivative is f'''(x) = e^x(2x + 3) + 6e^x(x + 3).

Q: Compute the second derivative of f(x) = ln(x).
A: The first derivative is f'(x) = 1/x, and the second derivative is f''(x) = -1/x^2.

Q: Find the second derivative of f(x) = tan(x).
A: The first derivative is f'(x) = sec^2(x), and the second derivative is f''(x) = 2sec^2(x)tan(x).

Q: Differentiate the function f(x) = x^4 + 6x^3 - 2x^2 + x.
A: The first derivative is f'(x) = 4x^3 + 18x^2 - 4x + 1, and the second derivative is f''(x) = 12x^2 + 36x - 4.

Q: Find the third derivative of f(x) = cos(2x).
A: The first derivative is f'(x) = -2sin(2x), the second derivative is f''(x) = -4cos(2x), and the third derivative is f'''(x) = 8sin(2x).

Q: Find the second derivative of f(x) = x^5 - 3x^4 + 2x^3 - x^2.
A: The first derivative is f'(x) = 5x^4 - 12x^3 + 6x^2 - 2x, and the second derivative is f''(x) = 20x^3 - 36x^2 + 12x - 2.

Q: Differentiate the function f(x) = 5x^2 + 3x + 7.
A: The first derivative is f'(x) = 10x + 3, and the second derivative is f''(x) = 10.

Q: Find the third derivative of f(x) = ln(x^2 + 1).
A: The first derivative is f'(x) = 2x/(x^2 + 1), the second derivative is f''(x) = (2(x^2 + 1) - 4x^2)/(x^2 + 1)^2, and the third derivative is f'''(x) = (12x(x^2 + 1) - 4x^3)/(x^2 + 1)^3.

Q: Compute the second derivative of f(x) = e^(-x^2).
A: The first derivative is f'(x) = -2xe^(-x^2), and the second derivative is f''(x) = (4x^2 - 2)e^(-x^2).

Q: Find the second derivative of f(x) = sin(x^2).
A: The first derivative is f'(x) = 2xcos(x^2), and the second derivative is f''(x) = 2cos(x^2) - 4x^2sin(x^2).

Q: Differentiate the function f(x) = 3x^2 + 4x + 1.
A: The first derivative is f'(x) = 6x + 4, and the second derivative is f''(x) = 6.

Q: Find the third derivative of f(x) = x^2 * sin(x).
A: The first derivative is f'(x) = 2xsin(x) + x^2cos(x), the second derivative is f''(x) = 2sin(x) + 4xcos(x) - x^2sin(x), and the third derivative is f'''(x) = 4cos(x) - 6xsin(x) - x^2cos(x).

Q: Compute the second derivative of f(x) = x * e^x.
A: The first derivative is f'(x) = e^x + xe^x, and the second derivative is f''(x) = 2e^x + xe^x.

Q: Find the second derivative of f(x) = cos(x) * sin(x).
A: The first derivative is f'(x) = cos^2(x) - sin^2(x), and the second derivative is f''(x) = -2cos(x)sin(x).

Q: Differentiate the function f(x) = x^3 - 3x^2 + 5x - 2.
A: The first derivative is f'(x) = 3x^2 - 6x + 5, and the second derivative is f''(x) = 6x - 6.

Q: Find the third derivative of f(x) = x * ln(x).
A: The first derivative is f'(x) = ln(x) + 1, the second derivative is f''(x) = 1/x, and the third derivative is f'''(x) = -1/x^2.

Q: Compute the second derivative of f(x) = e^x * cos(x).
A: The first derivative is f'(x) = e^x * (cos(x) - sin(x)), and the second derivative is f''(x) = e^x * (2cos(x) - sin(x)).

Q: Find the second derivative of f(x) = tan(x).
A: The first derivative is f'(x) = sec^2(x), and the second derivative is f''(x) = 2sec^2(x)tan(x).

Q: Differentiate the function f(x) = e^(3x).
A: The first derivative is f'(x) = 3e^(3x), and the second derivative is f''(x) = 9e^(3x).

Q: Find the third derivative of f(x) = x^4 * cos(x).
A: The first derivative is f'(x) = 4x^3cos(x) - x^4sin(x), the second derivative is f''(x) = 12x^2cos(x) - 8x^3sin(x) - x^4cos(x), and the third derivative is f'''(x) = 24xcos(x) - 24x^2sin(x) - 3x^4sin(x).

Q: Compute the second derivative of f(x) = x^2 * cos(x).
A: The first derivative is f'(x) = 2xcos(x) - x^2sin(x), and the second derivative is f''(x) = 2cos(x) - 4xsin(x) - x^2cos(x).

Q: Find the second derivative of f(x) = e^(x^2).
A: The first derivative is f'(x) = 2xe^(x^2), and the second derivative is f''(x) = (4x^2 + 2)e^(x^2).

Q: Differentiate the function f(x) = x^6 + 2x^4 - x^2 + 4.
A: The first derivative is f'(x) = 6x^5 + 8x^3 - 2x, and the second derivative is f''(x) = 30x^4 + 24x^2 - 2.

Q: Find the second derivative of f(x) = e^(x + 1).
A: The first derivative is f'(x) = e^(x + 1), and the second derivative is f''(x) = e^(x + 1).

Q: Compute the third derivative of f(x) = cos(3x).
A: The first derivative is f'(x) = -3sin(3x), the second derivative is f''(x) = -9cos(3x), and the third derivative is f'''(x) = 27sin(3x).

Q: Find the second derivative of f(x) = x^3 * cos(x).
A: The first derivative is f'(x) = 3x^2cos(x) - x^3sin(x), and the second derivative is f''(x) = 6xcos(x) - 6x^2sin(x) - x^3cos(x).

Q: Differentiate the function f(x) = sin(3x).
A: The first derivative is f'(x) = 3cos(3x), and the second derivative is f''(x) = -9sin(3x).

Q: Find the second derivative of f(x) = x * cos(x) + sin(x).
A: The first derivative is f'(x) = cos(x) - xsin(x) + cos(x), and the second derivative is f''(x) = -2sin(x) - xcos(x).

Best Indian Books on Successive Differentiation Problems with Solutions

Higher Engineering Mathematics – B.S. Grewal, Khanna Publishers
- Content: This book contains comprehensive coverage of successive differentiation problems, along with detailed solutions. The exercises focus on the differentiation of various functions and provide step-by-step breakdowns for clarity.
Advanced Engineering Mathematics – Erwin Kreyszig, Wiley India Pvt. Ltd.
- Content: This book includes both theoretical concepts and practical problems. It offers numerous worked examples for successive differentiation and helps develop the necessary skills for solving higher-level calculus problems.
Mathematics for Class 11 & 12 – R.D. Sharma, Dhanpat Rai & Co.
- Content: Focuses on the basics of calculus and includes numerous problems for successive differentiation. It’s suitable for high school students and provides both direct problems and application-based problems.
Mathematics for Class 12 (Part-1) – R.D. Sharma, Dhanpat Rai & Co.
- Content: The book provides successive differentiation problems relevant to the CBSE syllabus and provides ample practice problems to prepare for exams.
Problems in Calculus of One Variable – I.A. Maron, Shruti Publications
- Content: This book is a go-to resource for calculus learners, providing more than 500 problems on successive differentiation with solutions. It emphasizes understanding the application of derivatives in solving real-life problems.
Integral Calculus for JEE Main & Advanced – Amit M Agarwal, Arihant Publications
- Content: Geared towards students preparing for competitive exams like JEE, this book offers advanced problems on successive differentiation with detailed explanations and solutions to help crack higher-level questions.
Differential Calculus for Class 12 – S.K. Goyal, Arihant Publications
- Content: This book offers systematic explanations of the principles of successive differentiation, including multiple examples, with a focus on algebraic, trigonometric, and logarithmic functions.
Problems in Calculus of Variations – J. L. Roses, Springer India
- Content: The focus of this book is more on the calculus of variations but includes sections on successive differentiation, providing detailed examples and solutions that are applicable for engineering and applied mathematics students.
Calculus and Analytical Geometry – S. L. Loney, S. Chand Publishing
- Content: This classic book covers various aspects of calculus, including successive differentiation. It is famous for its clear and simple explanations, making it perfect for beginner and intermediate learners.
Mathematics for Class 12 – P. N. Gupta, R. S. Aggarwal, S. Chand Publishing

Content: An ideal resource for students tackling higher-level calculus in school exams, offering problems with solutions related to successive differentiation and differentiation of complex functions.

Introduction to Calculus – P.K. Jain & K. Jain, Wiley Eastern Ltd.

Content: This book is targeted at students learning calculus for the first time, with simple and concise problems, followed by solutions that build an understanding of successive differentiation.

Advanced Problems in Calculus of One Variable – V. S. Agrawal, Tata McGraw Hill Education

Content: Designed for those preparing for competitive exams, this book offers advanced problems involving successive differentiation with detailed step-by-step solutions.

Problems in Differential Calculus – A. R. Vasishta, Krishna Prakashan Media

Content: This book includes a wide array of problems on successive differentiation. Solutions are well-explained, with a special emphasis on derivatives of trigonometric and exponential functions.

Higher Mathematics for Engineering and Science – M. D. Raisinghania, S. Chand Publishing

Content: This book goes in-depth into various calculus topics, including successive differentiation. It provides an extensive collection of solved problems with practical applications.

Objective Mathematics – R.D. Sharma, Dhanpat Rai Publications

Content: This book covers objective problems based on successive differentiation with solutions. The problems are designed to help students grasp the theory and application of differentiation techniques.

A Textbook of Calculus – M. L. Khanna, Vikas Publishing House

Content: This book offers a mix of theory and practice problems, including challenging problems on successive differentiation, making it an excellent tool for engineering students.

Mathematical Methods for Physics and Engineering – K.F. Riley, Cambridge University Press

Content: A detailed resource that includes problems on calculus and successive differentiation, tailored for physics and engineering students who need to apply the concepts in real-world scenarios.

Problems in Differential Calculus – M. S. Tyagi, Central Publishing House

Content: Contains both elementary and complex problems, offering solutions to a wide variety of calculus questions, including successive differentiation.

Mathematics for Engineering Students – V.P. Gupta, Ane Books India

Content: This book is ideal for students looking to strengthen their understanding of calculus, featuring a section devoted entirely to successive differentiation problems.

Advanced Problems in Calculus – P.C. Taneja, Pustak Mahal

Content: This book provides a collection of advanced problems on successive differentiation, with explanations and solutions aimed at students pursuing higher studies in mathematics.

Successive differentiation is an important concept in calculus, often used to find the nth derivative of a function. For students and professionals alike, mastering this topic is essential, as it aids in solving real-world problems, especially in fields like engineering, physics, and economics. By understanding successive differentiation, one can find the rate of change of rates of change, which is useful for analyzing motion, growth, and other dynamic systems.

In successive differentiation, we begin by differentiating a function once, and then continue differentiating the result repeatedly. Each differentiation gives us a new function that represents the rate of change of the previous one. The process continues until we reach the desired number of derivatives.

For example, consider the function $f(x) = e^{x^2}$ . The first derivative will be $f'(x) = 2xe^{x^2}$ , and the second derivative will be $f''(x) = 2e^{x^2} + 4x^2e^{x^2}$ . With each step, the complexity of the expression increases, and the use of chain and product rules becomes crucial.

In terms of problems, textbooks focusing on this subject will usually cover a range of exercises starting from basic functions like polynomials, exponential, and logarithmic functions. For instance, problems involving polynomials often ask for the nth derivative, requiring the student to apply power rules and recognize patterns. For more advanced problems, students may encounter trigonometric and exponential functions, which introduce the need for the chain rule.

The key to mastering successive differentiation lies in practicing problems that test your ability to apply differentiation rules repeatedly and correctly. The exercises typically start with simple functions and gradually increase in difficulty, encouraging students to master the application of the product and chain rules in various scenarios.

Books that offer solutions to these problems are invaluable. By working through the solutions, students can develop their problem-solving skills and gain a deeper understanding of how differentiation techniques apply in different contexts. Each solution typically includes a step-by-step breakdown, ensuring that the learner not only reaches the correct answer but also understands the reasoning behind each step.

Moreover, successive differentiation problems are often part of larger topics such as the Taylor series and the analysis of curves. As students progress, they will encounter problems that require them to apply higher-level concepts such as series expansions and approximation techniques.

The problems vary in complexity, from basic exercises to advanced ones designed for competitive exams like JEE and other engineering entrance tests. For example, questions may involve differentiating a function multiple times, calculating higher-order derivatives, or applying these derivatives in real-world contexts such as optimization problems or curve sketching.

In conclusion, successive differentiation is a core skill in calculus that requires practice and conceptual clarity. With the right resources and consistent effort, students can master this topic and apply it effectively in their academic and professional careers. The books mentioned above provide a wealth of practice problems and solutions, helping students build their proficiency in successive differentiation.