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Aus Online Mathematik Brückenkurs 2

We can see the expression as "ln of something",

ln

where "something" is lnx.

Because we have a compound expression, we use the chain rule and obtain, roughly speaking, the outer derivative multiplied by the inner derivative,

ddxlnlnx=1lnxlnx

where the first factor on the right-hand side 1lnx is the outer derivative of lnlnx and the other factor lnx  is the inner derivative. Thus, we get

ddxlnlnx=1lnxx1=1xlnx.