If y=x^x, find d y d x - Vidyadip

Question

If $y=x^x,$ find $\frac{d y}{d x}$

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Answer

$y = x^x.$
Taking logarithm of both sides, we get
$\log y = \log (x^x)$
$\therefore \log y = x \log x$
Differentiating both sides $w.r.t.x,$ we get
$\frac{1}{y} \cdot \frac{d y}{d x}=x \cdot \frac{d}{d x}(\log x)+\log x \cdot \frac{d}{d x}(x)$
$=x \cdot \frac{1}{x}+\log x(1)$
$\therefore \frac{1}{y} \cdot \frac{ dy }{ dx }=1+\log x$
$\therefore \frac{ dy }{ dx }= y (1+\log x )$
$\therefore \frac{ dy }{ dx }= x ^{ x }(1+\log x )$

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