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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $fraction numerator cos space straight x over denominator log space straight x end fraction$ , x>0

Solution

Let space space space space space straight y equals fraction numerator cos space straight x over denominator log space straight x end fraction therefore space dy over dx equals fraction numerator left parenthesis log space straight x right parenthesis. begin display style straight d over dx end style left parenthesis cos space straight x right parenthesis minus cos space straight x. begin display style straight d over dx end style left parenthesis log space straight x right parenthesis over denominator left parenthesis log space straight x right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator left parenthesis log space straight x right parenthesis. left parenthesis negative sin space straight x right parenthesis minus cos space straight x. begin display style 1 over straight x end style over denominator left parenthesis log space straight x right parenthesis squared end fraction equals fraction numerator sin space straight x space log space straight x minus cos space straight x over denominator left parenthesis log space straight x right parenthesis squared end fraction

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