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Continuity And Differentiability

Question

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Find the second order derivatives of the following functions : log (log x)

Solution

Let space space space space space space straight y equals log left parenthesis log space straight x right parenthesis therefore space space dy over dx equals fraction numerator 1 over denominator log space straight x end fraction.1 over straight x equals fraction numerator 1 over denominator straight x space log space straight x end fraction therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator left parenthesis straight x space log space straight x right parenthesis. begin display style straight d over dx end style left parenthesis 1 right parenthesis minus 1. begin display style straight d over dx end style left parenthesis straight x space log space straight x right parenthesis over denominator left parenthesis straight x space log space straight x right parenthesis squared end fraction space space space space space space space space space space space space space space equals fraction numerator left parenthesis straight x space log space straight x right parenthesis.0 minus 1. open parentheses straight x. begin display style 1 over straight x end style plus log space straight x.1 close parentheses over denominator left parenthesis straight x space log space straight x right parenthesis squared end fraction equals fraction numerator 0 minus left parenthesis 1 plus log space straight x right parenthesis over denominator left parenthesis straight x space log space straight x right parenthesis squared end fraction space space space space space space space space space space space space space space equals negative fraction numerator left parenthesis 1 plus log space straight x right parenthesis over denominator left parenthesis straight x space log space straight x right parenthesis squared end fraction

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