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Continuity and Differentiability

Short Answer Type

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

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

tan left parenthesis straight x plus straight y right parenthesis plus tan left parenthesis straight x minus straight y right parenthesis equals 1

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353. Differentiate the following functions w.r.t.x:
$log square root of straight x squared plus straight y squared end root equals tan to the power of negative 1 end exponent straight y over straight x$

Here space space space log open parentheses straight x squared plus straight y squared close parentheses to the power of 1 half end exponent equals tan to the power of negative 1 end exponent straight y over straight x therefore space 1 half log left parenthesis straight x squared plus straight y squared right parenthesis equals tan to the power of negative 1 end exponent straight y over straight x Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma fraction numerator 1 over denominator 2 left parenthesis straight x squared plus straight y squared right parenthesis end fraction open parentheses 2 straight x plus 2 straight y dy over dx close parentheses equals fraction numerator 1 over denominator 1 plus begin display style straight y squared over straight x squared end style end fraction. open parentheses fraction numerator straight x begin display style dy over dx end style minus straight y.1 over denominator straight x squared end fraction close parentheses therefore space fraction numerator straight x plus straight y begin display style dy over dx end style over denominator straight x squared plus straight y squared end fraction equals fraction numerator straight x begin display style dy over dx end style minus straight y over denominator straight x squared plus straight y squared end fraction therefore space space space space straight x plus straight y dy over dx equals straight x dy over dx minus straight y therefore space left parenthesis straight x minus straight y right parenthesis dy over dx equals straight x plus straight y therefore space space space space space space space space space space space space dy over dx equals fraction numerator straight x plus straight y over denominator straight x minus straight y end fraction

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

straight x cubed plus straight y cubed equals sin left parenthesis straight x plus straight y right parenthesis

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355.

Find space dy over dx when space straight x equals straight a left parenthesis straight theta plus sinθ right parenthesis comma space straight y equals straight a left parenthesis 1 minus cosθ right parenthesis

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356.

Find space dy over dx space when space straight x equals straight a left parenthesis 1 minus cosθ right parenthesis comma space straight y equals straight a left parenthesis straight theta plus sinθ right parenthesis

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357.

Find space dy over dx when space straight x equals straight a space sinθ comma space straight y equals straight a open parentheses cosθ plus log space tan straight theta over 2 close parentheses

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358.

Find space dy over dx space when space straight x equals cos cubed straight theta comma space straight y equals straight b space sin cubed straight theta

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359.

If space space straight x equals straight a left parenthesis 1 minus cosθ right parenthesis comma space straight y equals straight a left parenthesis 1 plus sinθ right parenthesis comma space find dy over dx comma

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360.

Find space dy over dx when space straight x equals straight a left parenthesis straight theta plus sinθ right parenthesis comma space straight y equals straight a left parenthesis 1 plus cosθ right parenthesis.

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