Question Number 137280 by mathlove last updated on 31/Mar/21
Answered by EDWIN88 last updated on 31/Mar/21
$$\Leftrightarrow\:\mathrm{y}\:=\:\mathrm{x}^{\left(\frac{\pi}{\mathrm{ln}\:\mathrm{x}}\right)} =\:\mathrm{x}^{\pi\mathrm{log}\:_{\mathrm{x}} \left(\mathrm{e}\right)} \\ $$$$\Leftrightarrow\:\mathrm{y}=\:\mathrm{x}^{\mathrm{log}\:_{\mathrm{x}} \left(\mathrm{e}^{\pi} \right)} \:;\:\mathrm{y}\:=\:\mathrm{e}^{\pi} \:\Rightarrow\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{0}. \\ $$$$ \\ $$
Answered by mr W last updated on 31/Mar/21
$${y}={x}^{\frac{\pi}{\mathrm{ln}\:{x}}} \\ $$$$\mathrm{ln}\:{y}=\frac{\pi}{\mathrm{ln}\:{x}}×\mathrm{ln}\:{x}=\pi \\ $$$$\Rightarrow{y}={e}^{\pi} \\ $$$$\Rightarrow\frac{{dy}}{{dx}}=\mathrm{0} \\ $$