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
Advanced Engineering Mathematics – Erwin Kreyszig, Wiley India Pvt. Ltd.
Mathematics for Class 11 & 12 – R.D. Sharma, Dhanpat Rai & Co.
Mathematics for Class 12 (Part-1) – R.D. Sharma, Dhanpat Rai & Co.
Problems in Calculus of One Variable – I.A. Maron, Shruti Publications
Integral Calculus for JEE Main & Advanced – Amit M Agarwal, Arihant Publications
Differential Calculus for Class 12 – S.K. Goyal, Arihant Publications
Problems in Calculus of Variations – J. L. Roses, Springer India
Calculus and Analytical Geometry – S. L. Loney, S. Chand Publishing
Mathematics for Class 12 – P. N. Gupta, R. S. Aggarwal, S. Chand Publishing
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)=ex2f(x) = e^{x^2}. The first derivative will be f′(x)=2xex2f'(x) = 2xe^{x^2}, and the second derivative will be f′′(x)=2ex2+4x2ex2f''(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.
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