If y = x^x, dy/dx will be: Option 1: x^x Option 2: x^x(1 + x) Option 3: x^(x-1) Option 4: x^(x+1)

Answer: x^x(1 + x). Solution: y = x^x gives y = x x. Differentiating: (1/y)(dy/dx) = x + 1. Therefore dy/dx = x^x(1 + x).

Calculus

Continuity & Differentiability

Easy

common

If $y = x^x$ , $\frac{dy}{dx}$ will be:

$x^x$

$x^x(1 + \log x)$

$x^{x-1}$

$x^{x+1}$

✅ Correct Option: 2

$y = x^x$ gives $\ln y = x \ln x$ . Differentiating: $(1/y)(dy/dx) = \ln x + 1$ . Therefore $dy/dx = x^x(1 + \log x)$ .

2026: 19 May Shift 2

If $f(x+y) = f(x) + f(y)$ for all $x, y \in R$ (set of real numbers) and $f(x)$ is continuous at $x = 0$ , then identify the correct statement ?

2026: 25 May Shift 1

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. $f(x) = \log_e(x^2+1)$	I. not continuous at all integral values
B. $f(x) = \lvert x\rvert$	II. is continuous as well as differentiable at $x=0$
C. $f(x) = x^2$	III. Continuous everywhere
D. $f(x) = [x]$ , $[x]$ is greatest integer function	IV. Continuous everywhere but not differentiable at $x=0$

Choose the correct answer from the options given below:

2025: 30 May Shift 1

Let $y = \cos(\sin x^2)$ , then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is equal to

2022: 30 Aug Shift 1

Derivative of $x^3 + 1$ with respect to $x^2 + 1$ is

2025: 30 May Shift 2

$f(x) = \begin{cases}\frac{1- \tan x}{4x-\pi}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4}\end{cases}$

is continuous at $x = \frac{\pi}{4}$ , then the value of k is

2025: 14 May Shift 1

Let $y = \sin(\cos x^2)$ , then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is equal to