If $x = 3 \,tan \,t $ and $y = 3 \,sec \,t$, then the value of $\frac{d^2 y}{dx^2}$ at $t = \frac{\pi}{4}$, is :
- $\frac{3}{2 \sqrt{2}}$
- $\frac{1}{3 \sqrt{2}}$
- $\frac{1}{6}$
- $\frac{1}{6 \sqrt{2} }$
The Correct Option is D
Solution and Explanation
$ \frac{dy}{dt} = 3 \sec t \tan t $
$ \frac{dy}{dx} = \frac{\tan t}{\sec t} = \sin t $
$ \frac{d^{2}y}{dx^{2}} =\cos t \frac{dt}{dx} $
$= \frac{\cos t}{3 \sec^{2} t} = \frac{\cos^{3}t}{3} = \frac{1}{3.2\sqrt{2}} = \frac{1}{6\sqrt{2}} $
Top Questions on Differentiability
- Consider the quadratic equation \( ax^2 + bx + c = 0 \), where \( 2a + 3b + 6c = 0 \) and let \[ g(x) = \frac{a x^3}{3} + \frac{b x^2}{2} + c x \] Statement-I: The given quadratic equation \( ax^2 + bx + c = 0 \) has at least one root in \( (0, 1) \). Statement-II: Rolle's theorem is applicable to \( g(x) \) on \( [0, 1] \). Then:
- AP EAPCET - 2025
- Mathematics
- Differentiability
- If \(f(x) = \sec^{-1} \left(\frac{1}{2x^2 -1}\right)\) and \(g(x) = \tan^{-1} \left(\frac{\sqrt{1 + x^2} - 1}{x}\right)\), then the derivative of \(f(x)\) with respect to \(g(x)\) is?
- AP EAPCET - 2025
- Mathematics
- Differentiability
- If \( x = \sqrt{2}e^t(\sin t - \cos t) \) and \( y = \sqrt{2}e^t(\sin t + \cos t) \), then \( \left[ \frac{d^2y}{dx^2} \right]_{t=\frac{\pi}{4}} \) is:
- AP EAPCET - 2025
- Mathematics
- Differentiability
- If \( y = \tan^{-1}\left(\frac{x}{1+2x^2}\right) + \tan^{-1}\left(\frac{x}{1+6x^2}\right) \), then \( \frac{dy}{dx} = \)
- AP EAPCET - 2025
- Mathematics
- Differentiability
- If \( x = \sqrt{2}e^t(\sin t - \cos t) \) and \( y = \sqrt{2}e^t(\sin t + \cos t) \), then \( \left[ \frac{d^2y}{dx^2} \right]_{t=\frac{\pi}{4}} \) is:
- AP EAPCET - 2025
- Mathematics
- Differentiability
Questions Asked in JEE Main exam
- A force of 49 N acts tangentially at the highest point of a sphere (solid of mass 20 kg) kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is:
- JEE Main - 2025
- Quantum Mechanics
Let \( A = \{-3, -2, -1, 0, 1, 2, 3\} \). A relation \( R \) is defined such that \( xRy \) if \( y = \max(x, 1) \). The number of elements required to make it reflexive is \( l \), the number of elements required to make it symmetric is \( m \), and the number of elements in the relation \( R \) is \( n \). Then the value of \( l + m + n \) is equal to:
- JEE Main - 2025
- Sets and Relations
For hydrogen-like species, which of the following graphs provides the most appropriate representation of \( E \) vs \( Z \) plot for a constant \( n \)?
[E : Energy of the stationary state, Z : atomic number, n = principal quantum number]- JEE Main - 2025
- Hydrogen
- Concentrated nitric acid is labelled as 75%by mass. The volume in mL of the solution which contains 30 g of nitric acid is: Given: Density of nitric acid solution is 1.25 g/mL.
- JEE Main - 2025
- Solutions
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is $ 4 \_\_\_\_\_$.
- JEE Main - 2025
- permutations and combinations
JEE Main Notification
Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.