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Continuity and Differentiability

Short Answer Type

291. Differentiate the following w.r.t.x:4cosxlog_ex+5

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292. Differentiate the following w.r.t.x:
cos (a cos x + b sin x), for some constants a and b.

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293. Differentiate (x² + 7 x + 2) (e^x - sin x) w.r.t.x.

Let space space space space space straight y equals left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis therefore space dy over dx equals straight d over dx left square bracket left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis right square bracket space space space space space space space space space space space space space equals left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis straight d over dx left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis plus left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis straight d over dx left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis space space space space space space space space space space space space space equals left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis left parenthesis 2 straight x plus 7 right parenthesis space space space space space space space space space space space space space equals straight x squared left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus 7 straight x left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus 2 left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus 2 straight x left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis plus 7 left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis space space space space space space space space space space space space space equals straight x squared left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus straight x left parenthesis 9 straight e to the power of straight x minus 7 cos space straight x minus 2 sin space straight x right parenthesis plus 9 straight e to the power of straight x minus 2 cos space straight x minus 7 sin space straight x.

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294. Differentiate sin³ x + cos⁶ x w.r.t.x.

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295. Differentiate cos x³. sin² (x⁵) w.r.t.x.

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

fraction numerator straight e to the power of straight x over denominator sin space straight x end fraction

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

fraction numerator cos space straight x over denominator log space straight x end fraction

, x>0

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

fraction numerator sin squared straight x over denominator 1 plus cos squared space straight x end fraction

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

fraction numerator straight e to the power of ax over denominator sin left parenthesis bx plus straight c right parenthesis end fraction

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

fraction numerator sin space straight x plus log space straight x over denominator straight x to the power of 7 plus straight e to the power of straight x end fraction