Question

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

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Solution

Let space space straight y equals cos 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 space straight x equals tan space straight theta therefore space space space space straight y equals cos to the power of negative 1 end exponent open parentheses fraction numerator 2 tanθ over denominator 1 plus tan squared space straight theta end fraction close parentheses equals cos to the power of negative 1 end exponent open parentheses sin space 2 straight theta close parentheses equals cos to the power of negative 1 end exponent open square brackets cos open parentheses straight pi over 2 minus 2 straight theta close parentheses close square brackets space space space space space space space space space space equals straight pi over 2 minus 2 straight theta equals straight pi over 2 minus 2 space tan to the power of negative 1 end exponent straight x therefore space dy over dx equals 0 minus fraction numerator 2 over denominator 1 plus straight x squared end fraction

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