Question

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

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Solution

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

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