Question

$If space straight y equals sin open square brackets 2 space tan to the power of negative 1 end exponent square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction end root close square brackets comma space prove space that space dy over dx equals negative fraction numerator straight x over denominator square root of 1 minus straight x squared end root end fraction$

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Solution

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

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