find-d-dx-y-where-y-x-x-

Question Number 11552 by Nayon last updated on 28/Mar/17

f i n d \frac{d}{d x} (y) w h e r e y = \overset{\sqrt{x}}{} \sqrt{\sqrt{x}}

Answered by ajfour last updated on 28/Mar/17

y = {(\sqrt{x})}^{\frac{1}{\sqrt{x}}}

\ln y = \frac{\ln \sqrt{x}}{\sqrt{x}}

\frac{1}{y} \frac{d y}{d x} = (\frac{(\frac{1}{\sqrt{x}}) (\frac{1}{2 \sqrt{x}}) \sqrt{x} - \frac{\ln \sqrt{x}}{2 \sqrt{x}}}{x})

\frac{d y}{d x} = y (\frac{1 - \ln \sqrt{x}}{2 x \sqrt{x}})

\frac{d y}{d x} = \frac{{(\sqrt{x})}^{\frac{1}{\sqrt{x}}} (1 - \ln \sqrt{x})}{2 x \sqrt{x}} .

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