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Continuity And Differentiability

Question
CBSEENMA12035142
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Prove space that space straight d over dx left square bracket 2 straight x space tan to the power of negative 1 end exponent straight x minus log left parenthesis 1 plus straight x squared right parenthesis right square bracket equals 2 space tan to the power of negative 1 end exponent straight x.

Solution
straight L. straight H. straight S. space equals straight d over dx left square bracket 2 straight x space tan to the power of negative 1 end exponent straight x minus log left parenthesis 1 plus straight x squared right parenthesis right square bracket space space space space space space space space space space space space space equals straight d over dx left parenthesis 2 straight x space tan to the power of negative 1 end exponent straight x right parenthesis minus straight d over dx left square bracket log left parenthesis 1 plus straight x squared right parenthesis right square bracket space space space space space space space space space space space space space equals 2 straight d over dx left parenthesis straight x space tan to the power of negative 1 end exponent straight x right parenthesis minus fraction numerator 1 over denominator 1 plus straight x squared end fraction. straight d over dx left parenthesis 1 plus straight x squared right parenthesis space space space space space space space space space space space space space equals 2 open square brackets straight x straight d over dx left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis plus space tan to the power of negative 1 end exponent straight x. straight d over dx left parenthesis straight x right parenthesis close square brackets minus fraction numerator 1 over denominator 1 plus straight x squared end fraction.2 straight x equals 2 open square brackets straight x. fraction numerator 1 over denominator 1 plus straight x squared end fraction plus space tan to the power of negative 1 end exponent straight x.1 close square brackets minus fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction equals fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction plus 2 space tan to the power of negative 1 end exponent straight x minus fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction equals 2 space tan to the power of negative 1 end exponent straight x equals straight R. straight H. straight S.

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