Question

$Differentiate space tan to the power of negative 1 end exponent open parentheses fraction numerator straight x over denominator 1 plus square root of 1 minus straight x squared end root end fraction close parentheses space straight w. straight r. straight t. space sin open parentheses 2 space cot to the power of negative 1 end exponent space square root of fraction numerator 1 plus straight x over denominator 1 minus straight x end fraction end root close parentheses$

Solution

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

Also space straight u equals sin open square brackets 2 space cot to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 plus straight x over denominator 1 minus straight x end fraction end root close parentheses close square brackets Put space space space space straight x equals cos space straight theta therefore space space space space space space straight u equals sin open square brackets 2 space cot to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 plus cos space straight theta over denominator 1 minus cos space straight theta end fraction end root close parentheses close square brackets space space space space space space space space space space space space equals sin open square brackets 2 space cot to the power of negative 1 end exponent open parentheses square root of fraction numerator 2 space cos 2 begin display style straight theta over 2 end style over denominator 2 space sin 2 begin display style straight theta over 2 end style end fraction end root close parentheses close square brackets space space space space space space space space space space space space equals sin open square brackets 2 space cot to the power of negative 1 end exponent open parentheses cot straight theta over 2 close parentheses close square brackets space space space space space space space space space space space space equals sin open square brackets 2 cross times straight theta over 2 close square brackets space space space space space space space space space space space space equals sin space straight theta equals square root of 1 minus cos squared space straight theta end root therefore space straight u equals square root of 1 minus straight x squared end root therefore space du over dx equals fraction numerator negative 2 straight x over denominator square root of 1 minus straight x squared end root end fraction therefore space du over dx equals negative fraction numerator 2 straight x over denominator square root of 1 minus straight x squared end root end fraction Now space dy over du equals fraction numerator begin display style dy over dx end style over denominator begin display style du over dx end style end fraction equals fraction numerator 1 over denominator 2 square root of 1 minus straight x squared end root end fraction cross times fraction numerator square root of 1 minus straight x squared end root over denominator negative straight x end fraction equals negative fraction numerator 1 over denominator 2 straight x end fraction

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