题目
已知函数=ln [ ln (ln x)] ，则=ln [ ln (ln x)] _____.=ln [ ln (ln x)] =ln [ ln (ln x)] =ln [ ln (ln x)] =ln [ ln (ln x)]
已知函数
，则
_____.
题目解答
答案
对函数求导，设
，
，则
，
将
代入，则
故答案选
解析
步骤 1：求导
对函数$y=\ln [ \ln (\ln x)] $求导，设$u=\ln x$，$a=\ln u$，则$y=\ln a$，$y'(x)=y'(a)\cdot a'(u)\cdot u'(x)$
步骤 2：计算导数
$y'(x)=\dfrac {1}{a}\cdot \dfrac {1}{u}\cdot \dfrac {1}{x}$
$=\dfrac {1}{(\ln \ln x)(\ln x)x}$
步骤 3：代入$x={e}^{2}$
将${e}^{2}=x$代入，则$y'({e}^{2})=\dfrac {1}{(\ln \ln {e}^{2})(\ln {e}^{2}){e}^{2}}$
$=\dfrac {1}{2{e}^{2}\ln 2}$
对函数$y=\ln [ \ln (\ln x)] $求导，设$u=\ln x$，$a=\ln u$，则$y=\ln a$，$y'(x)=y'(a)\cdot a'(u)\cdot u'(x)$
步骤 2：计算导数
$y'(x)=\dfrac {1}{a}\cdot \dfrac {1}{u}\cdot \dfrac {1}{x}$
$=\dfrac {1}{(\ln \ln x)(\ln x)x}$
步骤 3：代入$x={e}^{2}$
将${e}^{2}=x$代入，则$y'({e}^{2})=\dfrac {1}{(\ln \ln {e}^{2})(\ln {e}^{2}){e}^{2}}$
$=\dfrac {1}{2{e}^{2}\ln 2}$