Ответ:

[tex]$ (\ln x)^{\ln x} * \left(\cfrac{\ln (\ln x)}{x} + \cfrac{1}{x}\right)[/tex]

Пошаговое объяснение:

[tex]$ (\ln x)^{\ln x} = e ^ {ln((\ln x)^{\ln x})} = e ^ {\ln x \ln (\ln x)}}[/tex]

[tex]((\ln x)^{\ln x})' = (e ^ {\ln x \ln (\ln x)}})' = e ^ {\ln x \ln (\ln x)}} * (\ln x \ln (\ln x))' = \\= (\ln x)^{\ln x} * \left(\cfrac{\ln (\ln x)}{x} + \ln x (\ln (\ln x))'\right) = \left[(\ln (\ln x))' = \cfrac{(\ln x)'}{\ln x} = \cfrac{1}{x\ln x} \right] = \\= (\ln x)^{\ln x} * \left(\cfrac{\ln (\ln x)}{x} + \cfrac{\ln x}{x\ln x}\right) = (\ln x)^{\ln x} * \left(\cfrac{\ln (\ln x)}{x} + \cfrac{1}{x}\right)[/tex]