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321.

If space straight y equals sin open square brackets square root of sin square root of straight x end root close square brackets comma space where space straight x greater than 0 comma space find space dy over dx

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322. If y = a sin x + b cos x, prove that

straight y squared plus open parentheses dy over dx close parentheses squared equals straight a squared plus straight b squared

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323.

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

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324.

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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325.

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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326.

If space straight y equals fraction numerator sin space straight x over denominator 1 plus begin display style fraction numerator cos space straight x over denominator 1 plus begin display style fraction numerator sin space straight x over denominator 1 plus begin display style fraction numerator cos space straight x over denominator 1 plus.... infinity end fraction end style end fraction end style end fraction end style end fraction comma space prov e space that space dy over dx equals fraction numerator left parenthesis 1 plus straight y right parenthesis cos space straight x plus straight y space sin space straight x over denominator 1 plus 2 straight y plus cos space straight x minus sin space straight x end fraction

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327. Differentiate the following w.r.t.x:

square root of fraction numerator 1 minus cos space straight x over denominator 1 plus cos space straight x end fraction end root

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328. Differentiate the following w.r.t.x:

square root of fraction numerator sec minus 1 over denominator sec plus 1 end fraction end root

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Let space space space space space straight y equals fraction numerator sin space straight x over denominator 1 plus cos space straight x end fraction equals fraction numerator 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style over denominator 2 cos squared begin display style straight x over 2 end style end fraction equals fraction numerator sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction equals tan straight x over 2 therefore space dy over dx equals sec squared straight x over 2. straight d over dx open parentheses straight x over 2 close parentheses equals sec squared straight x over 2 cross times 1 half equals 1 half sec squared straight x over 2

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330. Differentiate the following w.r.t.x:

fraction numerator 1 minus cos space straight x over denominator sin space straight x end fraction

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