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Continuity and Differentiability

Short Answer Type

341. If cosy = x cos (a + y), then prove that

dy over dx equals fraction numerator cos squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction

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342.

If space straight x space straight y plus straight y squared equals tan space straight x plus straight y comma space find dy over dx

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343. $Find space dy over dx space when space sin left parenthesis straight x space straight y right parenthesis plus straight x over straight y equals straight x squared minus straight y.$

space space space space space sin left parenthesis straight x space straight y right parenthesis plus straight x over straight y equals straight x squared minus straight y. Differentiating space both space sides space straight w. straight r. straight t. space straight x comma cos left parenthesis straight x space straight y right parenthesis. straight d over dx left parenthesis straight x space straight y right parenthesis plus fraction numerator straight y begin display style straight d over dx end style left parenthesis straight x right parenthesis minus straight x begin display style straight d over dx end style left parenthesis straight y right parenthesis over denominator straight y squared end fraction equals 2 straight x minus dy over dx therefore space cos left parenthesis straight x space straight y right parenthesis cross times open square brackets straight x dy over dx plus straight y.1 close square brackets plus fraction numerator straight y.1 minus straight x begin display style dy over dx end style over denominator straight y squared end fraction equals 2 straight x minus dy over dx therefore space straight x space straight y squared cos left parenthesis straight x space straight y right parenthesis dy over dx plus straight y cubed cos left parenthesis straight x space straight y right parenthesis plus straight y minus straight x dy over dx equals 2 straight x space straight y squared minus straight y squared dy over dx therefore space left square bracket straight x space straight y squared cos left parenthesis straight x space straight y right parenthesis minus straight x plus straight y squared right square bracket dy over dx equals 2 space straight x space straight y squared minus straight y cubed space cos left parenthesis straight x space straight y right parenthesis minus straight y therefore space dy over dx equals fraction numerator straight y left square bracket 2 space straight x space straight y minus straight y squared space cos left parenthesis straight x space straight y right parenthesis minus 1 right square bracket over denominator straight x space straight y squared cos left parenthesis straight x space straight y right parenthesis minus straight x plus straight y squared end fraction.

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344.

FInd space dy over dx of space 2 straight x plus 3 straight y equals sin space straight x.

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345.

FInd space dy over dx space of space 2 straight x plus 3 straight y equals sin space straight y

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346.

Find space dy over dx space of space straight a space straight x plus straight b space straight y squared equals cos space straight y space

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347.

Find space dy over dx space of space straight x space straight y plus straight y squared space equals tan space straight x plus straight y

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348.

Find space dy over dx space of space sin squared space straight y plus cos space straight x space straight y equals straight pi

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349. Differentiate the following functions w.r.t.x: sin²x+cos²y=1

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

sin space straight y plus log space straight y equals straight x squared plus 18 straight x plus 3

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