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Continuity And Differentiability

Question

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

Solution

Let space space space space space straight y equals sin left parenthesis log space straight x right parenthesis. therefore space dy over dx equals cos left parenthesis log space straight x right parenthesis.1 over straight x space space fraction numerator straight d squared straight y over denominator dx to the power of 2 straight x end exponent end fraction equals cos left parenthesis log space straight x right parenthesis. straight d over dx open parentheses 1 over straight x close parentheses plus 1 over straight x. straight d over straight d left square bracket cos left parenthesis log space straight x right parenthesis right square bracket space space space space space space space space space space space space space equals cos left parenthesis log space straight x right parenthesis. open parentheses negative 1 over straight x squared close parentheses plus 1 over straight x. open square brackets fraction numerator negative sin left parenthesis log space straight x right parenthesis over denominator straight x end fraction close square brackets space space space space space space space space space space space space space equals negative fraction numerator cos left parenthesis log space straight x right parenthesis over denominator straight x squared end fraction minus fraction numerator sin left parenthesis log space straight x right parenthesis over denominator straight x squared end fraction equals negative 1 over straight x squared left square bracket cos left parenthesis log space straight x right parenthesis plus sin left parenthesis log space straight x right parenthesis right square bracket

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