Question

Differentiate the following functions w.r.t.x: $fraction numerator straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x over denominator square root of 1 plus straight x squared end root end fraction$

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Solution

Let space space space space space straight y equals fraction numerator straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x over denominator square root of 1 plus straight x squared end root end fraction therefore space dy over dx equals fraction numerator square root of 1 plus straight x squared end root. begin display style straight d over dx end style left parenthesis straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x right parenthesis minus straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x. begin display style straight d over dx end style open parentheses square root of 1 plus straight x squared end root close parentheses over denominator open parentheses square root of 1 plus straight x squared end root close parentheses squared end fraction space space space space space space space space space space space space space equals fraction numerator square root of 1 plus straight x squared end root open parentheses straight e to the power of straight x squared end exponent space begin display style fraction numerator 1 over denominator 1 plus straight x squared end fraction end style plus straight e to the power of straight x squared end exponent.2 straight x. tan to the power of negative 1 end exponent straight x close parentheses minus straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x. begin display style fraction numerator 2 straight x over denominator 2 square root of 1 plus straight x squared end root end fraction end style over denominator 1 plus straight x squared end fraction space space space space space space space space space space space space space equals fraction numerator left parenthesis 1 plus straight x squared right parenthesis open parentheses begin display style fraction numerator straight e to the power of straight x squared end exponent over denominator 1 plus straight x squared end fraction end style plus 2 straight x space straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x close parentheses minus xe to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x over denominator left parenthesis 1 plus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction space space space space space space space space space space space space space equals fraction numerator straight e to the power of straight x squared end exponent plus 2 straight x left parenthesis 1 plus straight x squared right parenthesis straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x minus xe to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x over denominator left parenthesis 1 plus straight x squared right parenthesis to the power of 3 over 2 end exponent end fraction

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Continuity And Differentiability

Differentiate the following functions w.r.t.x: $fraction numerator straight e to the power of straight x squared end exponent space tan to the power of negative 1 end exponent straight x over denominator square root of 1 plus straight x squared end root end fraction$

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