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Q.

$\frac{d}{d x} \{{(x^{x})}^{x}\} =$

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a

${(x^{x})}^{x} {x (1 + \log x)}$

b

${(x^{x})}^{x} \cdot x (1 - 2 \log x)$

c

${(x^{x})}^{x} (1 + 2 \log x) x$

d

${(x^{x})}^{x} \cdot x^{2} (1 - 2 \log x)$

answer is C.

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Detailed Solution

$y = {(x^{x})}^{x} \Rightarrow \log y = x \log (x^{x})$

$\log y = x^{2} \log x$

$differentiarte w.r.t.x$

$\frac{1}{y} \times \frac{d y}{d x} = x^{2} (\frac{1}{x}) + (\log x) 2 x$

$\frac{d y}{d x} = {(x^{x})}^{x} (x + 2 x \log x)$

$\frac{d y}{d x} = {(x^{x})}^{x} (1 + 2 \log x) x$

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