MCQ
If $y = {\log _{\sin \,x}}\left( {\tan \,x} \right)$ , then ${\left( {\frac{{dy}}{{dx}}} \right)_{\pi /4}}$ is equal to
- A$\frac{4}{{\ln \,2}}$
- B$-4\ ln\ 2$
- ✓$\frac{{ - 4}}{{\ln \,2}}$
- D$4\ ln\ 2$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{(\log \sin \mathrm{x})\left(\frac{\sec ^{2} \mathrm{x}}{\tan \mathrm{x}}\right)-(\log \tan \mathrm{x})(\cot \mathrm{x})}{(\log \sin \mathrm{x})^{2}}$
or ${\left( {\frac{{dy}}{{dx}}} \right)_{\pi /4}} = \frac{{ - 4}}{{\log 2}}$ (On simplification)
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$f(x)=\min \{x-[x], 1+[x]-x\}$
where $[\mathrm{x}]$ is the greatest integer less than or equal to $\mathrm{x}$. Let $\mathrm{P}$ denote the set containing all $x \in[0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x \in(0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $\mathrm{P}$ and $\mathrm{Q}$ is equal to $......$