The derivative of y = x^ x is - Vidyadip

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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