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Continuity and Differentiability

Short Answer Type

311. Differentiate the following functions w.r.t.x:

log open parentheses square root of fraction numerator 1 minus cos space straight x over denominator 1 plus cos space straight x end fraction end root close parentheses

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312. Differentiate :

log space sin square root of straight x squared plus 1 end root

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313. $If space straight y equals log square root of fraction numerator 1 plus cos squared straight x over denominator 1 minus straight e squared straight x end fraction end root comma space find space dy over dx$

Here space straight y equals log open parentheses If space straight y equals log fraction numerator 1 plus cos squared straight x over denominator 1 minus straight e squared straight x end fraction close parentheses to the power of 1 half end exponent therefore space space space space space space straight y equals 1 half log open parentheses fraction numerator 1 plus cos squared straight x over denominator 1 minus straight e squared straight x end fraction close parentheses rightwards double arrow space space space space space space straight y equals 1 half left square bracket log left parenthesis 1 plus cos to the power of straight x right parenthesis minus log left parenthesis 1 minus straight e to the power of 2 straight x end exponent right parenthesis right square bracket therefore space dy over dx equals 1 half open square brackets fraction numerator 1 over denominator 1 plus cos squared straight x end fraction left parenthesis negative 2 cos space straight x space sin space straight x right parenthesis minus fraction numerator 1 over denominator 1 minus straight e to the power of 2 straight x end exponent end fraction left parenthesis negative 2 straight e to the power of 2 straight x end exponent right parenthesis close square brackets space space space space space space space space space space space space space equals 1 half open square brackets fraction numerator negative 2 sin space xcos space straight x over denominator 1 plus cos space straight x end fraction plus fraction numerator 2 straight e 2 straight x over denominator 1 minus straight e to the power of 2 straight x end exponent end fraction close square brackets therefore space dy over dx equals negative fraction numerator sin space xcos space straight x over denominator 1 plus cos squared straight x end fraction plus fraction numerator straight e to the power of 2 straight x end exponent over denominator 1 minus straight e to the power of 2 straight x end exponent end fraction

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314.

If space equals log square root of fraction numerator 1 plus sin squared straight x over denominator 1 minus tan space straight x end fraction end root comma space find space dy over dx.

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315. Differentiate the following functions w.r.t.x:

square root of 1 plus tan space straight x end root

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316. Differentiate the following functions w.r.t.x:

square root of 1 plus cot space straight x end root

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317. Differentiate the following functions w.r.t.x:

sin space 2 straight x. cos space 3 straight x

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318. Differentiate the following functions w.r.t.x:

straight x space cot cubed straight x

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319. Differentiate the following functions w.r.t.x:

straight e to the power of negative straight x squared end exponent. sin left parenthesis log space straight x right parenthesis

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320. Differentiate the following functions w.r.t.x:

straight e to the power of sin space straight x end exponent. sine to the power of straight x

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