MCQ
If $y = {x^{{x^{x......\infty }}}}$, then ${{dy} \over {dx}} = $
  • A
    ${{{y^2}} \over {x(1 + y\log x)}}$
  • ${{{y^2}} \over {x(1 - y\log x)}}$
  • C
    ${y \over {x(1 + y\log x)}}$
  • D
    ${y \over {x(1 - y\log x)}}$

Answer

Correct option: B.
${{{y^2}} \over {x(1 - y\log x)}}$
b
(b) $y = {x^{{x^{x.......\infty }}}}$ ==>$y = {x^y}$==>$\log y = y\log x$

Therefore, on differentiating $\frac{{dy}}{{dx}} = \frac{{{y^2}}}{{x(1 - y\log x)}}$.

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

Similar questions

If $A=\left[\begin{array}{cc}8 & 0 \\ 4 & -2 \\ 3 & 6\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & -2 \\ 4 & 2 \\ -5 & 1\end{array}\right],$ then find the matrix $X$, such that $2 \mathrm{A}+3 \mathrm{X}=5 \mathrm{B}$.
$\int {x\sin x\ {{\sec }^3}\ x\,\,\,dx} $  equal to
If set $A$ contains $5$ elements and the set $B$ contains $6$ elements, then the number of one$-$one and onto mappings from $A$ to $B$ is:
Statement $-1$ : Any function $f (x)$ is even function, when $f (-x) = f (x)$ over its specified domain. 

Statement $-2$ : $f(x) = \frac{1}{{\sqrt {1 - {x^2}} }} + \left[ {\frac{{{x^2} + x + 1}}{4}} \right]$ , where $[.]$ is greatest integer function. Function $f(x)$ is even function

If ${\rm{f}}\left( x \right) = \int\limits_{^{{\pi ^2}/16}}^{{x^2}} {\frac{{\sin x\,\cdot\sin \sqrt \theta  }}{{1 + {{\cos }^2}\sqrt \theta  }}} .d\theta $ then the value of $f ‘$$\left( {\frac{\pi }{2}} \right)$ , is
If ${\Delta _1} = \left| {\begin{array}{*{20}{c}}
  {{b^5}{c^6}\left( {{c^3} - {b^3}} \right)}&{{a^4}{c^6}\left( {{a^3} - {c^3}} \right)}&{{a^4}{b^5}\left( {{b^3} - {a^3}} \right)} \\ 
  {{b^2}{c^3}\left( {{b^6} - {c^6}} \right)}&{a{c^3}\left( {{c^6} - {a^6}} \right)}&{a{b^2}\left( {{a^6} - {b^6}} \right)} \\ 
  {{b^2}{c^3}\left( {{c^3} - {b^3}} \right)}&{a{c^3}\left( {{a^3} - {c^3}} \right)}&{a{b^2}\left( {{b^3} - {a^3}} \right)} 
\end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}}
  a&{{b^2}}&{{c^3}} \\ 
  {{a^4}}&{{b^5}}&{{c^6}} \\ 
  {{a^7}}&{{b^8}}&{{c^9}} 
\end{array}} \right|$ then ${\Delta _1}{\Delta _2}$ is equal to
Let $[t]$ denote the greatest integer less than or equal to $t$. Let $\mathrm{f}:[0, \infty) \rightarrow \mathrm{R}$ be a function defined by $f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which $\mathrm{f}$ is not continuous. Then $\sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to............
The equation of the plane parallel to the lines x - 1 = 2y - 5 = 2z and 3x = 4y - 11 = 3z -4 and passing through the point (2, 3, 3) is:
Which of the following differentials equation has y = x as one of its particular solution?
$\left| {\,\begin{array}{*{20}{c}}5&3&{ - 1}\\{ - 7}&x&{ - 3}\\9&6&{ - 2}\end{array}\,} \right| = 0$, then $ x$ is equal to