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Continuity And Differentiability

Question

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Differentiate the following functions w.r.t.x: $straight e to the power of negative straight x squared end exponent. sin left parenthesis log space straight x right parenthesis$

Solution

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

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