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Continuity And Differentiability

Question

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Differentiate x² e^3x w.r.t. (log x)².

Solution

Let space space straight y equals space straight x squared straight e to the power of 3 straight x end exponent space space and space straight u equals left parenthesis log space straight x right parenthesis squared Then space we space required space to space find space dy over du Now space dy over dx equals straight x squared. straight e to the power of 3 straight x end exponent.3 plus straight e to the power of 3 straight x end exponent.2 straight x equals straight e to the power of 3 straight x end exponent left square bracket 3 straight x squared plus 2 straight x right square bracket and space du over dx equals 2 left parenthesis log space straight x right parenthesis to the power of 1. straight d over dx left parenthesis log space straight x right parenthesis equals 2 space log.1 over straight x equals fraction numerator 2 space log space straight x over denominator straight x end fraction therefore space fraction numerator dy over denominator space space space du end fraction equals fraction numerator begin display style dy over dx end style over denominator begin display style du over dx end style end fraction equals straight e to the power of 3 straight x end exponent fraction numerator left parenthesis 3 straight x squared plus 2 straight x right parenthesis over denominator begin display style 2 over straight x log space straight x end style end fraction equals fraction numerator straight e to the power of 3 straight x end exponent left parenthesis 3 straight x squared plus 2 straight x squared right parenthesis over denominator 2 space log space straight x end fraction

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