Question

y=x^ln(x). найти y'(e)

Answer 1

$y = x^{\ln x}$

$\ln y = \ln x^{\ln x}$

$\ln y = \ln x \cdot \ln x$

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

$y = e^{\ln^{2}x}$

$y' = \left(e^{\ln^{2}x} \right)' = e^{\ln^{2}x} \cdot (\ln^{2}x)' = e^{\ln^{2}x} \cdot 2\ln x \cdot (\ln x)' = \dfrac{2\ln x \cdot e^{\ln^{2}x}}{x}$

$y'(e) = \dfrac{2\ln e \cdot e^{\ln^{2}e}}{e} = \dfrac{2e}{e} = 2$

Ответ: $y'(e) = 2$