Question

$If space straight y equals square root of sin space straight x plus square root of sin space straight x plus square root of sin space straight x plus.... infinity end root end root end root comma space prove space that space left parenthesis 2 straight y minus 1 right parenthesis dy over dx equals cos space straight x$

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Solution

SInce space straight y equals square root of sin space straight x plus square root of sin space straight x plus square root of sin space straight x plus.... infinity end root end root end root therefore space space space space space space space straight y equals square root of sin space straight x plus straight y end root Squaring comma space space space space space space space space space straight y squared equals sin space straight x plus straight y Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma 2 straight y dy over dx equals cos space straight x plus dy over dx comma space space space therefore space 2 straight y dy over dx minus dy over dx equals cos space straight x therefore space left parenthesis 2 straight y minus 1 right parenthesis dy over dx equals cos space straight x

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