MCQ
If $y = {x^{({x^x})}}$, then ${{dy} \over {dx}} = $
- A$y[{x^x}(\log ex).\log x + {x^x}]$
- B$y[{x^x}(\log ex).\log x + x]$
- ✓$y[{x^x}(\log ex).\log x + {x^{x - 1}}]$
- D$y[{x^x}({\log _e}x).\log x + {x^{x - 1}}]$
==> $\frac{1}{y}\frac{{dy}}{{dx}} = \frac{{dz}}{{dx}}.\log x + \frac{1}{x}.z$ , (where ${x^x} = z$)
$ \Rightarrow \frac{{dy}}{{dx}} = {x^{({x^x})}}\left[ {{x^x}(\log ex).\log x + {x^{x - 1}}} \right]$, $\left\{ \because \frac{dz}{dx}={{x}^{x}}\log ex \right\}$.
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$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1} ,\,\,\, x>0$
is continuous at $x=0$, then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to :
Let $\mathrm{R}$ be a relation on $\mathrm{A} \times \mathrm{B}$ define by $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ if and only if $3 \mathrm{ad}-7 \mathrm{bc}$ is an even integer. Then the relation $\mathrm{R}$ is