Question

$Find space dy over dx comma space when space straight y equals sin to the power of negative 1 end exponent straight x plus sin to the power of negative 1 end exponent square root of 1 minus straight x squared end root.$

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Solution

space space space space space space space space space space space straight y equals sin to the power of negative 1 end exponent straight x plus sin to the power of negative 1 end exponent square root of 1 minus straight x squared end root Put space space space space space straight x equals sin space straight theta space space space space space space space space space space space straight y equals sin to the power of negative 1 end exponent left parenthesis sin space straight theta right parenthesis plus sin to the power of negative 1 end exponent square root of 1 minus sin squared straight theta end root equals square root of 1 minus straight x squared end root equals sin to the power of negative 1 end exponent left parenthesis sin space straight theta right parenthesis plus sin to the power of negative 1 end exponent left parenthesis cos space straight theta right parenthesis space space space space space space space space space space space space space equals sin to the power of negative 1 end exponent left parenthesis sin space straight theta right parenthesis plus sin to the power of negative 1 end exponent open square brackets sin open parentheses straight x over 2 minus straight theta close parentheses close square brackets equals straight theta plus straight pi over 2 equals straight pi over 2 therefore space dy over dx equals 0

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