Successive Differentiation-Previous year questions

Previous Year Questions on Successive Differentiation

Successive Differentiation has been a fundamental concept in mathematics, enabling the computation of higher-order derivatives. Its historical significance lies in its application to solving complex mathematical models and real-world scenarios. The importance of Successive Differentiation is reflected in its wide range of applications in physics, engineering, and economics.

Vidyasagar University

2023-24 (NEP)

If \(y=\frac{x^{3}}{x^{2}-1} \) then find \(y_{n}\). [2]

2023-24 (CBCS)

If \(y=x^{n-1}\log{x}\), then prove that \(y_{n}=\frac{(n-1)!}{x} \). [2]
If \(y=\frac{x^{2}-6}{x^{3}-x^{2}-2x}\), find \(y_{n} \). [2]
Wrtie the Leibnitz theorem successive derivatives up to \(4^{th}\) order. [2]
If \(y=\sin{\left(m\cos^{-1}{\sqrt{x}} \right)} \) then prove that \(\lim\limits_{x\to 0}\frac{y_{n+1}}{y_{n}}=\frac{4n^{2}-m^{2}}{4n+2} \). [5]

2022-23 (CBCS)

2021-22 (CBCS)

If \(y=\sin{\left(m\cos^{-1}{\sqrt{x}} \right)} \) then prove that \(\lim\limits_{x\to 0}\frac{y_{n+1}}{y_{n}}=\frac{4n^{2}-m^{2}}{4n+2} \). [4]

2020-21 (CBCS)

If \(y=\left(\sin^{-1}{x} \right)^{2} \) show that \(\left(1-x^{2} \right)y_{n-2}\right)-\left(2n+1 \right)xy_{n-1}\right) -n^{2} y_{n}\right)=0\). Also find \(y_{n}(0) \). [6]

2019-20 (CBCS)

If \(y=e^{ax}\cos^{2}{bx} \), find \(y_{n}\) where \(a,b\gt 0\). [2]
If \(y=x^{n-1}\log{x} \), then prove that \(y_{n}=\frac{(n-1)!}{x}\). [2]
Let \(P_{n}=D^{n}\left(x^{n}\log{x} \right) \), hence show that \(P_{n}=n!\left(\log{x}+1+\frac{1}{2}+…+\frac{1}{n} \right) \). [5]

2018-19 (CBCS)

If \(y=2\cos{x}\left(\sin{x}-\cos{x} \right) \) then find the value of \(\left(b_{20}\right)_{0}\). [2]
If \(y=\sin{\left(m\cos^{-1}{\sqrt{x}} \right)} \) then prove that \(\lim\limits_{x\to 0}\frac{y_{n+1}}{y_{n}}=\frac{4n^{2}-m^{2}}{4n+2} \). [4]

2017-18 (CBCS)

Wrtie the Leibnitz theorem successive derivatives up to \(4^{th}\) order. [2]

FAQs

What is Successive Differentiation?
It is the process of finding higher-order derivatives of a function by repeatedly differentiating it.
Where is Successive Differentiation applied?
It is applied in physics, engineering, and economics for solving dynamic systems and modeling behavior.
What are higher-order derivatives?
These are derivatives obtained by differentiating a function multiple times.
Is Successive Differentiation part of university courses?
Yes, it is included in the curriculum of many universities in Differential Calculus.
What is the significance of higher-order derivatives?
They help in analyzing the concavity, inflection points, and behavior of functions.
What topics are related to Successive Differentiation?
Topics include Linear Algebra, Classical Algebra, and Abstract Algebra.
How is Successive Differentiation used in real life?
It is used in optimizing systems, analyzing motion, and modeling economic trends.
Are there limitations to Successive Differentiation?
Yes, it requires the function to be differentiable multiple times, which may not always be the case.
Where can I find practice questions on Successive Differentiation?
Explore Mathematics Questions or topics like Differential Calculus.
Can Successive Differentiation be automated?
Yes, software tools like Mathematica and MATLAB can perform it efficiently.