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Continuity And Differentiability

Question

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Differentiate the following function w.r.t.x: $straight e to the power of cot to the power of negative 1 end exponent straight x squared end exponent$

Solution

Let space space space space space straight y equals straight e to the power of cot to the power of negative 1 end exponent straight x squared end exponent therefore space dy over dx equals straight e to the power of cot to the power of negative 1 end exponent straight x squared end exponent. straight d over dx left parenthesis cot to the power of negative 1 end exponent space straight x squared right parenthesis equals straight e to the power of cot to the power of negative 1 end exponent straight x squared end exponent. fraction numerator negative 1 over denominator 1 plus left parenthesis straight x squared right parenthesis squared end fraction. straight d over dx left parenthesis straight x squared right parenthesis space space space space space space space space space space space space space equals straight e to the power of cot to the power of negative 1 end exponent straight x squared end exponent. fraction numerator negative 1 over denominator 1 plus straight x to the power of 4 end fraction.2 straight x equals negative fraction numerator 2 straight x over denominator 1 plus straight x to the power of 4 end fraction straight e to the power of cot to the power of negative 1 end exponent straight x squared end exponent

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