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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.

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

Let space space space space space straight y equals fraction numerator sin squared straight x over denominator 1 plus cos squared space straight x end fraction therefore space dy over dx equals fraction numerator left parenthesis 1 plus cos squared space straight x right parenthesis. begin display style straight d over dx end style left parenthesis sin squared straight x right parenthesis minus sin squared straight x. begin display style straight d over dx end style left parenthesis 1 plus cos squared space straight x right parenthesis over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator left parenthesis 1 plus cos squared space straight x right parenthesis. left parenthesis 2 sin space straight x space cos space straight x right parenthesis minus sin squared straight x. left parenthesis negative 2 cos space straight x space sin space straight x right parenthesis over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator 2 sin space xcos space straight x left square bracket 1 plus cos squared space straight x plus sin squared straight x right square bracket over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction equals fraction numerator sin space 2 straight x left square bracket 1 plus 1 right square bracket over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction equals fraction numerator 2 sin space 2 straight x over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared 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

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