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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $sin left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis$

Solution

Let space space space space space straight y equals sin left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis therefore space dy over dx equals cos left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis. straight d over dx left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis space space space space space space space space space space space space space equals cos left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis. fraction numerator 1 over denominator 1 plus left parenthesis straight e to the power of negative straight x end exponent right parenthesis squared end fraction. straight d over dx left parenthesis straight e to the power of negative straight x end exponent right parenthesis space space space space space space space space space space space space space equals cos left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis. fraction numerator 1 over denominator 1 plus left parenthesis straight e to the power of negative straight x end exponent right parenthesis squared end fraction. straight e to the power of negative straight x end exponent. straight d over dx left parenthesis negative straight x right parenthesis space space space space space space space space space space space space space equals cos left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis. fraction numerator 1 over denominator 1 plus left parenthesis straight e to the power of negative straight x end exponent right parenthesis squared end fraction. straight e to the power of negative straight x end exponent. left parenthesis negative 1 right parenthesis space space space space space space space space space space space space space equals negative fraction numerator straight e to the power of negative straight x end exponent over denominator 1 plus straight e to the power of negative 2 straight x end exponent end fraction. cos left parenthesis tan to the power of negative 1 end exponent straight e to the power of negative straight x end exponent right parenthesis.

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