The derivative of ^(-1)(2x/(1+x^2)) w.r.t. ^(-1)(2x/(1-x^2)) is Option 1: 1/2 Option 2: 2 Option 3: (1-x^2)/(1+x^2) Option 4: 1

Answer: 1. Solution: Let x = . Then 2x/(1+x^2) = 2, so ^(-1)( 2) = 2. And 2x/(1-x^2) = 2, so ^(-1)( 2) = 2. Both u and v equal 2, so du/dv = 1.

Calculus

Continuity & Differentiability

Medium

The derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ w.r.t. $\tan^{-1}\left(\frac{2x}{1-x^2}\right)$ is

$\frac{1}{2}$

$\frac{1-x^2}{1+x^2}$

✅ Correct Option: 4

Let $x = \tan\theta$ .

Then $\frac{2x}{1+x^2} = \sin 2\theta$ , so $\sin^{-1}(\sin 2\theta) = 2\theta$ .

And $\frac{2x}{1-x^2} = \tan 2\theta$ , so $\tan^{-1}(\tan 2\theta) = 2\theta$ .

Both $u$ and $v$ equal $2\theta$ , so $\frac{du}{dv} = 1$ .

2022: 7 Aug Shift 2

$f(x) = [x]$ , where $[\,]$ represents greatest integer function.

(A) For $2 \leq x < 3$ , $[x] = 3$

(B) For $2 \leq x < 3$ , $[x] = 2$

(D) Left hand derivative of f at $x = 2$ is zero

(E) $f(x)$ is not differentiable at $x = 2$

Choose the correct answer from the options given below :

2026: 19 May Shift 2

The derivative of $|x|$ with respect to $x$ , where $x \in R \sim \{0\}$ is

2025: 29 May Shift 2

If $xy = e^{(x-y)}$ , then $\frac{dy}{dx}$ is equal to:

2026: 25 May Shift 1

If $e^y(x+1) = 1$ , then $\dfrac{d^2y}{dx^2}$ is equal to

2024: 16 May Shift 1

Let $[x]$ denote the greatest integer function. Then match List-I with List-II:

List-I	List-II
(A) $\vert x - 1\vert + \vert x - 2\vert$	(I) is differentiable everywhere except at $x = 0$
(B) $x - \vert x\vert$	(II) is continuous everywhere
(C) $x - [x]$	(III) is not differentiable at $x = 1$
(D) $x \, \vert x\vert$	(IV) is differentiable at $x = 1$

2025: 3 June Shift 2

If $\sin y = x \cos(a + y)$ , then $\frac{dy}{dx}$ is equal to