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Continuity and Differentiability

Short Answer Type

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

fraction numerator straight e to the power of straight x over denominator 1 plus cos space straight x end fraction

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

straight e to the power of sin square root of straight x end exponent

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

straight e to the power of 3 straight x end exponent log left parenthesis sin space 2 straight x right parenthesis

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

straight e to the power of square root of 1 minus straight x squared end root end exponent

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305.

Differentiate space straight e to the power of straight x space log square root of straight x. tan space straight x space straight w. straight r. straight t. straight x.

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

sin left parenthesis log space straight x right parenthesis

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

straight e to the power of cos space straight x end exponent

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

log left parenthesis cos space 5 straight x right parenthesis

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309. Differentiate the following functions w.r.t.x: $log space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis$

Let space space space space space space straight y equals log space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis therefore space space dy over dx equals fraction numerator 1 over denominator sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis end fraction. straight d over dx left square bracket sin space left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis right square bracket space space space space space space space space space space space space space space equals fraction numerator 1 over denominator space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis end fraction. cos left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis. straight d over dx left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis space space space space space space space space space space space space space space equals fraction numerator 1 over denominator space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis end fraction. cos left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis. left parenthesis straight e to the power of straight x plus 5 right parenthesis space space space space space space space space space space space space space space equals left parenthesis straight e to the power of straight x plus 5 right parenthesis cot left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis

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

log space tan open parentheses straight pi over 4 plus straight x over 2 close parentheses

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