Question

найти производную функции $y=ln^{2x}x$

Answer 1

$y=\left(\ln^{2\,x}\left(x\right)\right)=\left({e}^{2\,x\,\ln\left(\ln\left(x\right)\right)}\right)\\y'={e}^{2\,x\,\ln\left(\ln\left(x\right)\right)}\cdot \left(2\,x\,\ln\left(\ln\left(x\right)\right)\right)'=\\=2\,{e}^{2\,x\,\ln\left(\ln\left(x\right)\right)}\cdot \left(\left(x\right)'\cdot \ln\left(\ln\left(x\right)\right)+\left(\ln\left(\ln\left(x\right)\right)\right)'\cdot x\right)=$$\\=2\,{e}^{2\,x\,\ln\left(\ln\left(x\right)\right)}\cdot \left(1\cdot \ln\left(\ln\left(x\right)\right)+\dfrac{1}{\ln\left(x\right)}\cdot \left(\ln\left(x\right)\right)'\cdot x\right)=\\=2\,\ln^{2\,x}\left(x\right)\,\left(\ln\left(\ln\left(x\right)\right)+\dfrac{1}{\ln\left(x\right)}\right)$