d-dx-x-x-please-give-the-answer-with-proof-

Question Number 11547 by Nayon last updated on 28/Mar/17

\frac{d}{d x} (x^{x}) = ? [p l e a s e g i v e t h e a n s w e r w i t h p r o o f]

Answered by sma3l2996 last updated on 28/Mar/17

x^{x} = e^{x l n (x)}

s o \frac{d (x^{x})}{d x} = \frac{d (e^{x l n (x)})}{d x} = \frac{d}{d x} (x l n (x)) e^{l n (x^{x})}

= (l n (x) + 1) x^{x}

Answered by ajfour last updated on 28/Mar/17

y = x^{x}

\ln y = x \ln x

\frac{1}{y} \frac{d y}{d x} = \ln x + 1

\frac{d y}{d x} = y (1 + \ln x)

\frac{d y}{d x} = x^{x} (1 + \ln x) .

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