If y = e^x x x^2 , then dy dx = - Vidyadip

MCQ

If $y = {{{e^x}\log x} \over {{x^2}}}$, then ${{dy} \over {dx}} = $

A
${{{e^x}[1 + (x + 2)\log x]} \over {{x^3}}}$
B
${{{e^x}[1 - (x - 2)\log x]} \over {{x^4}}}$
C
${{{e^x}[1 - (x - 2)\log x]} \over {{x^3}}}$
✓
${{{e^x}[1 + (x - 2)\log x]} \over {{x^3}}}$

✓

Answer

Correct option: D.

${{{e^x}[1 + (x - 2)\log x]} \over {{x^3}}}$

d
(d) $\frac{{dy}}{{dx}} = - 2{x^{ - 3}}{e^x}\log x + {x^{ - 2}}\left( {{e^x}\log x + \frac{{{e^x}}}{x}} \right)$

$ = {e^x}\left[ {\frac{{1 + (x - 2)\log x}}{{{x^3}}}} \right]$

Aliter: Taking $\log $, $\log y = x + \log \log x - 2\log x$

==> $\frac{1}{y}\frac{{dy}}{{dx}} = 1 + \frac{1}{{x\log x}} - \frac{2}{x}$

==> $\frac{{dy}}{{dx}} = \frac{{{e^x}\log x}}{{{x^2}}}$ $\left[ {\frac{{x\log x + 1 - 2\log x}}{{x\log x}}} \right]$

$ = \frac{{{e^x}[(x - 2)\log x + 1]}}{{{x^3}}}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 5. continuity and differentiation→STD 12 - 5. continuity and differentiation — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

The inverse of matrix $A = \left[ {\begin{array}{*{20}{c}}0&1&0\\1&0&0\\0&0&1\end{array}} \right]$ is

If $A $ and $ B$ are square matrices of order $3$ such that $|A| = - 1,$ $|B| = 3,$then $|3AB|$=

$\int_0^{\pi /2} {\frac{{d\theta }}{{1 + \tan \theta }}} = $

If a matrix $A$ is such that $3A^3 + 2A^2 + 5A + I = 0,$ then $A^{-1}$ equal to:

If $\overrightarrow{ a }+\overrightarrow{ b }=\hat{ i }$ and $\overrightarrow{ a }=2 \hat{ i }-2 \hat{ j }+2 \hat{ k }$, then $|\overrightarrow{ b }|$ equals:

The matrix $\begin{bmatrix} 5 & 10 & 3 \\ -2 & -4 & 6 \\ -1 & -2 & \text{b} \end{bmatrix}$ is a singular matrix, if the value of $b$ is:

Define a function $f: R \rightarrow R$ by $f(x)=\left\{\begin{array}{cc}\frac{\sin x^2}{x}, & \text { for } x<0 \\ x^2+a x+b, & \text { for } x \geq 0\end{array}\right.$ Suppose $f(x)$ is differentiable of $R$. Then,

$\text{f}(\text{x})=1+2\sin\text{x}+3\cos^{2}\text{x}, 0<\text{x}<\frac{2\pi}{3}$ is :

Let $f:(0, \infty) \rightarrow R$ be given by $f(x)=\int_{\frac{1}{x}}^x e^{-\left(t+\frac{1}{t}\right)} \frac{d t}{t}$. Then

$(A)$ $f(x)$ is monotonically increasing on $[1, \infty)$

$(B)$ $f(x)$ is monotonically decreasing on $(0,1)$

$(C)$ $f(x)+f\left(\frac{1}{x}\right)=0$, for all $x \in(0, \infty)$

$(D)$ $f\left(2^x\right)$ is an odd function of $x$ on $R$

$\cos^{-1}\Big(\frac{1}{2}\Big)$