Question:
Find the second order derivatives of the function
\(tan^{-1}x\)
Find the second order derivatives of the function
\(tan^{-1}x\)
\(tan^{-1}x\)
Updated On: Sep 12, 2023
Hide Solution
Verified By Collegedunia
Solution and Explanation
The correct answer is \(=\frac{-2x}{(1+x^2)^2}\)
Let \(y=tan^{-1}x\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^2}\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}[\frac{1}{1+x^2}]\)
\(=\frac{d}{dx}(1+x^2)^{-1}\)
\(=(-1).(1+x^2)^{-2}.\frac{d}{dx}(1+x^2)\)
\(=\frac{-1}{(1+x^2)^2}\times 2x\)
\(=\frac{-2x}{(1+x^2)^2}\)
Let \(y=tan^{-1}x\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^2}\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}[\frac{1}{1+x^2}]\)
\(=\frac{d}{dx}(1+x^2)^{-1}\)
\(=(-1).(1+x^2)^{-2}.\frac{d}{dx}(1+x^2)\)
\(=\frac{-1}{(1+x^2)^2}\times 2x\)
\(=\frac{-2x}{(1+x^2)^2}\)
Was this answer helpful?
0
0
Top Questions on Continuity and differentiability
- If $\tan^{-1} (x^2 - y^2) = a$, where $a$ is a constant, then $\frac{dy}{dx}$ is:
- CBSE CLASS XII - 2025
- Mathematics
- Continuity and differentiability
- If the function f(x) = $\begin{cases}\frac{k\cos x}{\pi - 2x} & ; x \neq \frac{\pi}{2} \\ 3 & ; x = \frac{\pi}{2} \end{cases}$ is continuous at x = $\frac{\pi}{2}$, then k is equal to
- CUET (UG) - 2025
- Mathematics
- Continuity and differentiability
- Let \( y=\sin(\cos(x^2)) \). Find \( \frac{dy}{dx} \) at \( x=\frac{\sqrt{\pi}}{2} \).
- CUET (UG) - 2025
- Mathematics
- Continuity and differentiability
Match List-I with List-II
List-I List-II (A) \( f(x) = |x| \) (I) Not differentiable at \( x = -2 \) only (B) \( f(x) = |x + 2| \) (II) Not differentiable at \( x = 0 \) only (C) \( f(x) = |x^2 - 4| \) (III) Not differentiable at \( x = 2 \) only (D) \( f(x) = |x - 2| \) (IV) Not differentiable at \( x = 2, -2 \) only
Choose the correct answer from the options given below:- CUET (UG) - 2025
- Mathematics
- Continuity and differentiability
Match List-I with List-II
List-I List-II (A) \( f(x) = |x| \) (I) Not differentiable at \( x = -2 \) only (B) \( f(x) = |x + 2| \) (II) Not differentiable at \( x = 0 \) only (C) \( f(x) = |x^2 - 4| \) (III) Not differentiable at \( x = 2 \) only (D) \( f(x) = |x - 2| \) (IV) Not differentiable at \( x = 2, -2 \) only
Choose the correct answer from the options given below:- CUET (UG) - 2025
- Mathematics
- Continuity and differentiability
View More Questions
Questions Asked in CBSE CLASS XII exam
- You are Susheel of 10, R.K. Marg, New Delhi. You come across a report stating that a number of people in cities suffer from respiratory diseases, lung cancer, and heart diseases due to air pollution. Write a letter to the editor of a national daily expressing your concern and suggestions and ways to improve the situation.
\includegraphics[width=0.75\linewidth]5b.JPG- CBSE CLASS XII - 2025
- Letter Writing
- Let \( A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}, B = \begin{bmatrix} -5 \\ -2 \end{bmatrix}, C = [9 \ \ 7], \) which of the following is defined?
- What is the total number of possible matrices of order $3 \times 3$ with each entry as $\sqrt{2}$ or $\sqrt{3}$?
- Your school is organizing a road safety awareness workshop for students of class IX – XII. As the head boy of your school, draft a notice informing the students about the workshop. Include other necessary details. You are Ashna/Ashish. Put your notice in a box.
- CBSE CLASS XII - 2025
- Letter Writing
- Write a letter to the editor of a local daily highlighting the need for better public transportation in your city. You may use the given cues along with your own ideas to draft the letter. You are Sakshi / Sant, a resident of 46, C Colony, Crown Nagar.
- Challenges – Overcrowding in buses, poor maintenance, insufficient coverage, inadequate infrastructure
- Congestion due to private small time transport modes, unpredictable delays.
- Solutions – Increase frequency, regular maintenance, upgrading infrastructure, affordable fare.
- CBSE CLASS XII - 2025
- Letter Writing
View More Questions
Concepts Used:
Second-Order Derivative
The Second-Order Derivative is the derivative of the first-order derivative of the stated (given) function. For instance, acceleration is the second-order derivative of the distance covered with regard to time and tells us the rate of change of velocity.
As well as the first-order derivative tells us about the slope of the tangent line to the graph of the given function, the second-order derivative explains the shape of the graph and its concavity.
The second-order derivative is shown using \(f’’(x)\text{ or }\frac{d^2y}{dx^2}\).