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Continuity and Differentiability

Short Answer Type

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

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

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

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

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

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

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

fraction numerator cot squared straight x minus 1 over denominator cot squared straight x plus 1 end fraction

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

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

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336.

If space straight y equals square root of fraction numerator 1 minus sin space 2 straight x over denominator 1 plus sin space 2 straight x end fraction end root comma space show space that space dy over dx plus sec squared open parentheses straight pi over 4 minus straight x close parentheses equals 0

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337. Differentiate :

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

w.r.t.x.

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338. $Given space that space cos space straight x over 2. cos space straight x over 2. cos straight x over 8.... equals fraction numerator sin space straight x over denominator straight x end fraction comma space prove space that 1 over 2 squared sec squared straight x over 2 plus 1 over 2 to the power of 4 sec squared straight x over 4 plus..... equals cosec squared straight x minus 1 over straight x squared$

We space have space space space space space space cos space straight x over 2. cos space straight x over 2. cos straight x over 8.... equals fraction numerator sin space straight x over denominator straight x end fraction therefore space log open square brackets cos space straight x over 2. cos space straight x over 2. cos straight x over 8.... close square brackets equals log open parentheses fraction numerator sin space straight x over denominator straight x end fraction close parentheses therefore space log open parentheses cos space straight x over 2 close parentheses plus log open parentheses cos space straight x over 2 close parentheses plus log open parentheses cos straight x over 8 close parentheses plus... equals log space sin space straight x minus log space straight x Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma space space space space fraction numerator negative begin display style 1 half end style sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction plus fraction numerator negative begin display style 1 fourth end style sin begin display style straight x over 4 end style over denominator cos begin display style straight x over 4 end style end fraction plus fraction numerator negative begin display style 1 over 8 end style sin begin display style straight x over 8 end style over denominator cos begin display style straight x over 8 end style end fraction plus... equals fraction numerator cos space straight x over denominator sin space straight x end fraction minus 1 over straight x or space space minus 1 half tan space straight x over 2 minus 1 fourth tan straight x over 4 minus 1 over 8 tan straight x over 8... equals negative cot space straight x minus 1 over straight x Again space differentiating space straight w. straight r. straight t. straight x comma we space get comma space space space space space minus 1 over 2 squared sec squared straight x over 2 minus 1 over 2 to the power of 4 sec squared straight x over 4 minus... equals negative cosec squared straight x plus 1 over straight x squared therefore space 1 over 2 squared sec squared straight x over 2 plus 1 over 2 to the power of 4 sec squared straight x over 4 plus... equals cosec squared straight x minus 1 over straight x squared

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339. If sin x = y sin (x + b), show that

dy over dx equals fraction numerator sin space straight b over denominator sin squared left parenthesis straight x plus straight b right parenthesis end fraction

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340. If sin y=x sin(a+y), then prove that

dy over dx equals fraction numerator sin squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction

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