MCQ
The derivative of $y = {x^{\ln x}}$ is
- A${x^{\ln x}}\ln x$
- B${x^{{\rm{ln}}\,x - 1}}{\rm{ln}}\,x$
- ✓$2{x^{\ln x - 1}}\ln \,x$
- D${x^{\ln x - 2}}$
==> $\frac{1}{y}\frac{{dy}}{{dx}} = \frac{{2\ln x}}{x}$
==> $\frac{{dy}}{{dx}} = y\frac{{2\ln x}}{x} = \frac{{2({x^{\ln x}})\ln x}}{x}$
==> $\frac{{dy}}{{dx}} = 2{x^{\ln x - 1}}\ln x$.
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Define $g(x)=\int_1^x f(t) d t, x \in(1, \infty)$. Let $\alpha$ denote the number of solutions of the equation $g(x)=0$ in the interval $(1,8]$ and $\beta=\lim _{x \rightarrow 1+} \frac{g(x)}{x-1}$. Then the value of $\alpha+\beta$ is equal to. . . . . .