Question

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

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Solution

Let space space straight y equals sin to the power of negative 1 end exponent open parentheses 2 straight x square root of 1 minus straight x squared end root close parentheses space semicolon space Put space straight x equals sin space straight theta therefore space space space space straight y equals sin to the power of negative 1 end exponent open parentheses 2 sin space straight theta square root of 1 minus sin squared space straight theta end root close parentheses equals sin to the power of negative 1 end exponent open parentheses 2 sin space straight theta space cos space straight theta 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 therefore space space space space straight y equals 2 sin to the power of negative 1 end exponent straight x space space space rightwards double arrow space dy over dx equals fraction numerator 2 over denominator square root of 1 minus straight x squared end root end fraction

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