Question

$Find space dy over dx space in space the space following space colon space sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses$

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Solution

Le 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 Put space straight x equals tan space straight theta therefore space 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 tan squared straight theta end fraction close parentheses equals sin to the power of negative 1 end exponent left parenthesis sin space 2 straight theta right parenthesis equals 2 straight theta space space space rightwards double arrow space 2 tan to the power of negative 1 end exponent straight x therefore space dy over dx equals fraction numerator 2 over denominator 1 plus straight x squared end fraction.

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