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Continuity and Differentiability

Short Answer Type

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

Let space space space space space straight y equals 4 cos space straight x space log subscript straight e straight x plus 5 therefore space dy over dx equals straight d over dx left square bracket 4 cos space straight x space log space straight x plus 5 right square bracket equals straight d over dx left parenthesis 4 space cos space straight x space log space straight x right parenthesis plus straight d over dx left parenthesis 5 right parenthesis space space space space space space space space space space space space space equals space 4 straight d over dx left parenthesis cos space straight x space log space straight x right parenthesis plus straight d over dx left parenthesis 5 right parenthesis space space space space space space space space space space space space space equals 4 open square brackets cos space straight x. straight d over dx left parenthesis log space straight x right parenthesis. left parenthesis space straight x right parenthesis plus log space straight x. straight d over dx left parenthesis left parenthesis cos space straight x right parenthesis close square brackets plus straight d over dx left parenthesis 5 right parenthesis space space space space space space space space space space space space space equals 4 open square brackets cos space straight x.1 over straight x log space straight x. left parenthesis negative sin space straight x right parenthesis close square brackets plus 0 equals 4 open square brackets fraction numerator cos space straight x over denominator straight x end fraction minus sin space straight x. log space straight x close square brackets

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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

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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