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Question Number 137280 by mathlove last updated on 31/Mar/21

Answered by EDWIN88 last updated on 31/Mar/21

$$\iff \mathrm{y}={\mathrm{x}}^{\left(\frac{\pi }{\ln \mathrm{x}}\right)}={\mathrm{x}}^{\pi \log {}_{\mathrm{x}}\left(\mathrm{e}\right)}$$$$\iff \mathrm{y}={\mathrm{x}}^{\log {}_{\mathrm{x}}\left({\mathrm{e}}^{\pi }\right)};\mathrm{y}={\mathrm{e}}^{\pi }\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=0.$$$$$$

Answered by mr W last updated on 31/Mar/21

$$y=x^{\frac{\pi }{\ln x}}$$$$\ln y=\frac{\pi }{\ln x}\times \ln x=\pi$$$$\Rightarrow y=e^{\pi }$$$$\Rightarrow \frac{dy}{dx}=0$$

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