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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

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

Let space space space space space straight y equals straight e to the power of sin space straight x end exponent. sine to the power of straight x therefore space dy over dx equals straight e to the power of sin space straight x end exponent. straight d over dx left square bracket sine to the power of straight x right square bracket plus sin space straight e to the power of straight x. straight d over dx left parenthesis straight e space to the power of sin space straight x end exponent right parenthesis space space space space space space space space space space space space space equals straight e to the power of sin space straight x end exponent. cose to the power of straight x straight d over dx left parenthesis straight e to the power of straight x right parenthesis plus sin space straight e to the power of straight x. straight e to the power of sin space straight x end exponent. straight d over dx left parenthesis sin space straight x right parenthesis space space space space space space space space space space space space space equals straight e to the power of sin space straight x end exponent. cos space straight e to the power of straight x. ex plus sin space straight e to the power of straight x. straight e to the power of sin space straight x end exponent. cos space straight x space space space space space space space space space space space space space equals straight e to the power of sin space straight x end exponent left square bracket straight e to the power of straight x space cos space straight e to the power of straight x plus sin space straight e to the power of straight x. cos space straight x right square bracket

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