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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
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
    ${x^{\ln x - 2}}$
✓

Answer

Correct option: C.
$2{x^{\ln x - 1}}\ln \,x$
c
(c) $y = {x^{\ln x}}$ ==> $\ln y = {(\ln x)^2}$

==> $\frac{1}{y}\frac{{dy}}{{dx}} = \frac{{2\ln x}}{x}$

==> $\frac{{dy}}{{dx}} = y\frac{{2\ln x}}{x} = \frac{{2({x^{\ln x}})\ln x}}{x}$

==> $\frac{{dy}}{{dx}} = 2{x^{\ln x - 1}}\ln x$.

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