Question

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

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Solution

Let space straight y equals tan to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 minus sin space straight x over denominator 1 plus sin space straight x end fraction end root close parentheses equals tan to the power of negative 1 end exponent open parentheses square root of fraction numerator cos squared begin display style straight x over 2 end style plus sin squared begin display style straight x over 2 end style minus 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style over denominator cos squared straight x over 2 plus sin squared straight x over 2 plus 2 sin straight x over 2 cos straight x over 2 end fraction end root close parentheses space space space space space space space space space equals tan to the power of negative 1 end exponent square root of open parentheses fraction numerator cos begin display style straight x over 2 end style plus sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style plus sin begin display style straight x over 2 end style end fraction close parentheses squared end root equals tan to the power of negative 1 end exponent open parentheses fraction numerator cos begin display style straight x over 2 end style minus sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style plus sin begin display style straight x over 2 end style end fraction close parentheses space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style fraction numerator cos begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction end style minus begin display style fraction numerator sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction end style over denominator begin display style fraction numerator cos begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction end style plus begin display style fraction numerator sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction end style end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 minus tan begin display style straight x over 2 end style over denominator 1 plus tan begin display style straight x over 2 end style end fraction close square brackets space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets tan open parentheses straight pi over 4 minus straight x over 2 close parentheses close square brackets equals straight pi over 2 minus 1 half. straight x therefore space dy over dx equals 0 minus 1 half cross times 1 equals negative 1 half

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Continuity And Differentiability

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

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