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Continuity And Differentiability

Question
CBSEENMA12035216
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If space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus straight x squared over denominator 1 minus straight x squared end fraction close parentheses comma space prove space that space dy over dx equals fraction numerator 4 over denominator 1 plus straight x squared end fraction

Solution
Here space space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus straight x squared over denominator 1 minus straight x squared end fraction close parentheses Put space space space space space straight x equals tan space straight theta therefore space space space space If space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 tan space straight theta over denominator 1 plus tan squared space straight theta end fraction close parentheses plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus tan squared space straight theta over denominator 1 minus tan squared space straight theta end fraction close parentheses or space space space space space space space space straight y equals sin to the power of negative 1 end exponent left parenthesis sin space 2 straight theta right parenthesis plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus tan squared space straight theta over denominator 1 minus tan squared straight theta end fraction close parentheses therefore space space space space space space space space straight y equals 2 space straight theta plus 2 space straight theta comma space or space straight y equals 4 space straight theta therefore space space space space space space space space straight y equals 4 space tan to the power of negative 1 end exponent straight x therefore dy over dx equals fraction numerator 4 over denominator 1 plus straight x squared end fraction

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