Question:
If for $x \epsilon \left(0, \frac{1}{4}\right) ,$ the derivative of $ \tan^{-1} \left(\frac{6x \sqrt{x}}{1-9x^{3}}\right) $ is $\sqrt{x} . g(x)$, then $g(x)$ equals :
If for $x \epsilon \left(0, \frac{1}{4}\right) ,$ the derivative of $ \tan^{-1} \left(\frac{6x \sqrt{x}}{1-9x^{3}}\right) $ is $\sqrt{x} . g(x)$, then $g(x)$ equals :
Updated On: Apr 5, 2026
- $\frac{3x \sqrt{x}}{1 - 9x^3}$
- $\frac{3x }{1 - 9x^3}$
- $\frac{3 }{1 + 9x^3}$
- $\frac{9 }{1 + 9x^3}$
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
$f \left(x\right) = 2 \tan^{-1}\left(3x\sqrt{x}\right)$
For x $\in\left(0, \frac{1}{4}\right)$
$f'\left(x\right) = \frac{9\sqrt{x}}{1+9x^{3}}$
$g\left(x\right) = \frac{9}{1+9x^{3}}$
For x $\in\left(0, \frac{1}{4}\right)$
$f'\left(x\right) = \frac{9\sqrt{x}}{1+9x^{3}}$
$g\left(x\right) = \frac{9}{1+9x^{3}}$
Was this answer helpful?
0
0
Top JEE Main Mathematics Questions
- Let the midpoints of the sides of a triangle \(ABC\) be \( \left(\frac{5}{2},7\right), \left(\frac{5}{2},3\right)\) and \( (4,5) \). If its incentre is \((h,k)\), then \(3h+k\) is equal to:
Find the area of the region \[ R = \{(x, y) : xy \le 27,\; 1 \le y \le x^2 \}. \]
- JEE Main - 2026
- Mathematics
- Integration and Area Calculation
- Let \(f(x) = \begin{cases} \frac{1}{3} & , x \le \frac{\pi}{2} \\ \frac{b(1 - \sin x)}{(\pi - 2x)^2} & , x>\frac{\pi}{2} \end{cases}\). If \(f\) is continuous at \(x = \frac{\pi}{2}\), then the value of \(\int_0^{3b-6} |x^2 + 2x - 3| \, dx\) is:
- Let a line \(L_1\) pass through the origin and be perpendicular to the lines \(L_2: \vec{r} = (3 + t)\hat{i} + (2t - 1)\hat{j} + (2t + 4)\hat{k}\) and \(L_3: \vec{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}\). If \((a, b, c)\), \(a \in \mathbb{Z}\), is the point on \(L_3\) at a distance of \(\sqrt{17}\) from the point of intersection of \(L_1\) and \(L_2\), then \((a + b + c)^2\) is equal to ________.
- JEE Main - 2026
- Mathematics
- Integration
- The sum of squares of all the real solutions of the equation \(\log_{(x+1)}(2x^2 + 5x + 3) = 4 - \log_{(2x+3)}(x^2 + 2x + 1)\) is equal to ________.
- JEE Main - 2026
- Mathematics
- Ellipse Geometry
View More Questions
Top JEE Main Differentiability Questions
- Let \(\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}\), \(\vec{b} = 10\hat{i} + 2\hat{j} - \hat{k}\) and a vector \(\vec{c}\) be such that \(2(\vec{a} \times \vec{b}) + 3(\vec{b} \times \vec{c}) = \vec{0}\). If \(\vec{a} \cdot \vec{c} = 15\), then \(\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})\) is equal to:
- JEE Main - 2026
- Mathematics
- Differentiability
- Let \[ f(x)= \begin{cases} x^3+8, & x<0,\\ x^2-4, & x\ge0, \end{cases} \qquad g(x)= \begin{cases} (x-8)^{1/3}, & x<0,\\ (x+4)^{1/2}, & x\ge0. \end{cases} \] Then the number of points where the function \(g\circ f\) is discontinuous is ______.
- JEE Main - 2026
- Mathematics
- Differentiability
- Let $f(x) = \begin{cases} e^{x-1}, & x<0 \\ x^2 - 5x + 6, & x \ge 0 \end{cases}$ and $g(x) = f(|x|) + |f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha + \beta$ is equal to _______.
- JEE Main - 2026
- Mathematics
- Differentiability
- Let $f(x)=|\log_e x|-|x-1|+5$. Statement 1: $f(x)$ is differentiable for all $x\in(0,\infty)$
Statement 2: $f(x)$ is increasing in $(1,\infty)$
Statement 3: $f(x)$ is decreasing in $(0,1)$
Which of the following is correct?
- JEE Main - 2026
- Mathematics
- Differentiability
- Let the function \(f(x) = (x^2 - 1)|x^2 - ax + 2| + \cos|x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to:
- JEE Main - 2025
- Mathematics
- Differentiability
View More Questions
Top JEE Main Questions
- Let the midpoints of the sides of a triangle \(ABC\) be \( \left(\frac{5}{2},7\right), \left(\frac{5}{2},3\right)\) and \( (4,5) \). If its incentre is \((h,k)\), then \(3h+k\) is equal to:
- JEE Main - 2026
- Geometry
- A thin half ring of radius $35 \text{ cm}$ is uniformly charged with a total charge of $Q$ coulomb. If the magnitude of the electric field at centre of the half ring is $100 \text{ V/m}$, then the value of $Q$ is ______ $\text{nC}$.
($\epsilon_o = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2$ and $\pi = 3.14$)- JEE Main - 2026
- Electric Field
Find the area of the region \[ R = \{(x, y) : xy \le 27,\; 1 \le y \le x^2 \}. \]
- JEE Main - 2026
- Integration and Area Calculation
- Solution A is prepared by dissolving 1 g of a protein (molar mass = 50000 g mol\(^{-1}\)) in 0.5 L of water at 300 K. Its osmotic pressure is \( x \) bar. Solution B is made by dissolving 2 g of the same protein in 1 L of water at 300 K. Osmotic pressure of solution B is \( y \) bar. Entire solution of A is mixed with entire solution of B at same temperature. The osmotic pressure of resultant solution is \( z \) bar. \( x \), \( y \), and \( z \) respectively are:
- JEE Main - 2026
- Raoult's Law and Colligative Properties
An object of uniform density rolls up the curved path with the initial velocity $v_o$ as shown in the figure. If the maximum height attained by an object is $\frac{7v_o^2}{10 g}$ ($g=$ acceleration due to gravity), the object is a _______
- JEE Main - 2026
- Rotational Motion and Torque
View More Questions
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.