Question

$Find space dy over dx space in space the space following space colon space tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight x minus straight x cubed over denominator 1 minus 3 straight x squared end fraction close parentheses comma space minus fraction numerator 1 over denominator square root of 3 end fraction less than straight x less than fraction numerator 1 over denominator square root of 3 end fraction$

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Solution

Let space space space space straight y equals space tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight x minus straight x cubed over denominator 1 minus 3 straight x squared end fraction close parentheses Put space space space space straight x equals tan space straight theta therefore space space space space space space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator 3 tan space straight theta minus tan cubed space straight theta over denominator 1 minus 3 tan squared space straight theta end fraction close parentheses equals space tan to the power of negative 1 end exponent left parenthesis tan space 3 straight theta right parenthesis equals 3 straight theta space space space space rightwards double arrow space straight y equals 3 space tan to the power of negative 1 end exponent space straight x therefore space dy over dx equals fraction numerator 3 over denominator 1 plus straight x squared end fraction.

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