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Continuity and Differentiability

Short Answer Type

341. If cosy = x cos (a + y), then prove that $dy over dx equals fraction numerator cos squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction$

because space space cos space straight y equals straight x space cos left parenthesis straight a plus straight y right parenthesis therefore space space space space space space space space space space straight x equals fraction numerator cos space straight y over denominator cos left parenthesis straight a plus straight y right parenthesis end fraction Differentiating space both space sides space straight w. straight r. straight t. space straight y comma space space space space space space space space space dx over dy equals fraction numerator cos left parenthesis straight a plus straight y right parenthesis. begin display style straight d over dy end style left parenthesis cos space straight y right parenthesis minus cos space straight y begin display style straight d over dy end style left square bracket cos left parenthesis straight a plus straight y right parenthesis right square bracket over denominator cos squared left parenthesis straight a plus straight y right parenthesis end fraction therefore space space space space space dy over dx equals negative fraction numerator cos left parenthesis straight a plus straight y right parenthesis sin space straight y plus cos space straight y space sin left parenthesis straight a plus straight y right parenthesis over denominator cos squared left parenthesis straight a plus straight y right parenthesis end fraction space space space space space space space space space space space space space space space space space equals fraction numerator sin left square bracket left parenthesis straight a plus straight y right parenthesis minus straight y right square bracket over denominator cos squared left parenthesis straight a plus straight y right parenthesis end fraction equals fraction numerator sin space straight a over denominator cos squared left parenthesis straight a plus straight y right parenthesis end fraction Now space dy over dx equals fraction numerator 1 over denominator begin display style dx over dy end style end fraction 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.

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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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