d/dx(e^(2_e x^3)) equals Option 1: 30x^4 Option 2: 6x^5 Option 3: 5x^6 Option 4: 3x^2

Answer: 6x^5. Solution: The expression e^(2_e x^3) can be simplified before differentiating. Using the logarithm power rule _e x^3 = 3_e x: 2_e x^3 = 2 x 3_e x = 6_e x The expression becomes: e^(6_e x) Rewriting 6_e x as _e x^6: e^(6_e x) = e^(_e x^6) Using the property that e^(_e a) = a (since e and _e are inverse functions): e^(_e x^6) = x^6 Applying the power rule d/dx(x^n) = nx^(n-1): d/dx(x^6) = 6x^5 Therefore, the answer is option 2: 6x^5

Calculus

Continuity & Differentiability

Medium

common

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

$30x^4$

$6x^5$

$5x^6$

$3x^2$

✅ Correct Option: 2

The expression $e^{2\log_e x^3}$ can be simplified before differentiating.

Using the logarithm power rule $\log_e x^3 = 3\log_e x$ :

$2\log_e x^3 = 2 \times 3\log_e x$

$= 6\log_e x$

The expression becomes:

$e^{6\log_e x}$

Rewriting $6\log_e x$ as $\log_e x^6$ :

$e^{6\log_e x} = e^{\log_e x^6}$

Using the property that $e^{\log_e a} = a$ (since $e$ and $\log_e$ are inverse functions):

$e^{\log_e x^6} = x^6$

Applying the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ :

$\frac{d}{dx}(x^6) = 6x^5$

Therefore, the answer is option 2: $6x^5$

2026: 19 May Shift 1

Match List-I with List-II

List-I	List-II
(A) $f(x) = \begin{cases} x^{10}-1 & , x \leq 1 \\ x^2 & , x > 1 \end{cases}$	(I) is continuous at all points of domain
(B) $g(x) = [x]$ , $[x]$ denotes the greatest integer function	(II) is continuous at $x = 0$
(C) $h(x) = \begin{cases} \dfrac{\sin x}{x} & , x < 0 \\ x+1 & , x \geq 0 \end{cases}$	(III) is discontinuous at $x = -2$
(D) $p(x) = \begin{cases} \dfrac{x}{\left\vert x \right\vert} & , x < 0 \\ -1 & , x \geq 0 \end{cases}$	(IV) is discontinuous at $x = 1$

Choose the correct answer from the options given below:

2026: 22 May Shift 1

Let a function $f(x)=\alpha+(\beta^2+5\beta+6)|x|+\gamma|x|^4$ , where $\alpha, \beta$ and $\gamma$ are real number. Then the function $f(x)$ is differentiable at $x=0$ if

2026: 29 May Shift 1

Match List-I with List-II

List-I (Function $y=$ )	List-II ( $\dfrac{dy}{dx}$ at $x=0$ )
(A) $x+e^x$	(I) 1
(B) $\tan^{-1}x-\dfrac{x}{2}$	(II) $\dfrac{1}{2}$
(C) $\tan^{-1}x$	(III) 0
(D) $x+\cos^{-1}x$	(IV) 2

Choose the correct answer from the options given below:

2026: 25 May Shift 1

The derivative of $x^{5x}$ with respect to $x$ is

2026: 31 May Shift 2

If the function $f(x) = \dfrac{(4^x-1)^3}{\sin\dfrac{x}{4}\cdot\log\left(1+\dfrac{x^2}{3}\right)}$ is continuous at $x = 0$ , then the value of $f(0)$ is: (Take $\log_e x = \log x$ )

2026: 19 May Shift 2

For the function $f(x) = -2|\log_e|x||$ , which of the following is true?

ddx(e2log⁡ex3)\frac{d}{dx}\left(e^{2\log_e x^3}\right)dxd​(e2loge​x3) equals

Solution

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$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals