Question

$Find space dy over dx space in space the space following space colon sin to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses comma space 0 less than straight x less than 1$

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Solution

Let space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses Put space straight x equals tan space straight theta therefore space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 1 minus tan squared straight theta over denominator 1 plus tan squared straight theta end fraction close parentheses equals sin to the power of negative 1 end exponent left parenthesis cos space 2 straight theta right parenthesis equals sin to the power of negative 1 end exponent open square brackets sin open parentheses straight pi over 2 minus 2 straight theta close parentheses close square brackets therefore space straight y 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 2. fraction numerator 1 over denominator 1 plus straight x squared end fraction

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