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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

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

because space space space sin space straight y equals straight x space sin left parenthesis straight a plus straight y right parenthesis therefore space space space space space space space space space space straight x equals fraction numerator sin space straight y over denominator sin left parenthesis straight a plus straight y right parenthesis end fraction Differentiating space both space sides space straight w. straight r. straight t. space straight y. space space space space space space space space space dx over dy equals fraction numerator sin left parenthesis straight a plus straight y right parenthesis begin display style straight d over dy end style left parenthesis sin space straight y right parenthesis minus sin space straight y begin display style straight d over dy end style left square bracket sin left parenthesis straight a plus straight y right parenthesis right square bracket over denominator sin squared left parenthesis straight a plus straight y right parenthesis end fraction space space space space space space space space space space space space space space space space space equals fraction numerator sin left parenthesis straight a plus straight y right parenthesis cos space straight y minus sin space straight y. cos left parenthesis straight a plus straight y right parenthesis over denominator sin squared left parenthesis straight a plus straight y right parenthesis end fraction equals fraction numerator sin left square bracket left parenthesis straight a plus straight y right parenthesis minus straight y right square bracket over denominator sin squared left parenthesis straight a plus straight y right parenthesis end fraction equals fraction numerator sin space straight a over denominator sin squared left parenthesis straight a plus straight y right parenthesis end fraction Now space dy over dx equals fraction numerator 1 over denominator begin display style dx over dy end style end fraction 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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