Question

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

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Solution

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

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