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Continuity and Differentiability

Short Answer Type

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

left parenthesis straight x space cos space straight x right parenthesis to the power of straight x plus left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent

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

straight e to the power of sin space straight x end exponent plus left parenthesis tan space straight x right parenthesis to the power of straight x

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

left parenthesis sin space straight x right parenthesis to the power of tan space space straight x end exponent plus left parenthesis cos space straight x right parenthesis to the power of sec space straight x end exponent

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394. Differentiate the following functions w.r.t. x : $straight x to the power of tan space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent$

Let space straight y equals straight x to the power of tan space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent Put space straight x to the power of tan space straight x end exponent equals straight u comma space left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent equals straight v therefore space straight y equals straight u plus straight v therefore space dy over dx equals du over dx plus dv over dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Now space straight u equals straight x to the power of tan space straight x end exponent therefore space log space straight u equals log space straight x to the power of tan space straight x end exponent therefore space log space straight u equals tan space straight x. log space straight x therefore space 1 over straight u du over dx equals fraction numerator tan space straight x space over denominator straight x end fraction plus log space straight x. sec squared straight x therefore space du over dx equals straight u open parentheses fraction numerator tan space straight x over denominator straight x end fraction plus log space straight x. sec squared straight x close parentheses therefore space du over dx equals straight x to the power of tan space straight x end exponent open parentheses fraction numerator tan space straight x over denominator straight x end fraction plus log space straight x. sec squared straight x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis and space straight v equals space left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent therefore space log space straight v equals space left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent therefore space log space straight v equals cos space straight x. log space sin space straight x therefore space 1 over straight v dv over dx equals cos space straight x. fraction numerator cos space straight x over denominator sin space straight x end fraction minus log space sin space straight x. sin space straight x therefore space dv over dx equals straight v left square bracket cos space straight x. cot space straight x minus sin space straight x. log space sin space straight x right square bracket therefore space dv over dx equals left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent left square bracket cos space straight x. cot space straight x minus sin space straight x. log space sin space straight x right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis From space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma we space get comma dy over dx equals straight x to the power of tan space straight x end exponent open parentheses fraction numerator tan space straight x over denominator straight x end fraction plus log space straight x. sec squared straight x close parentheses plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent left square bracket cos space straight x. cot space straight x minus sin space straight x. log space sin space straight x right square bracket

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

straight x to the power of straight x to the power of straight x end exponent plus straight x to the power of sin space straight x end exponent

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396. If x=sec ө-cos ө, y=secⁿө-cosⁿ ө then show that

left parenthesis straight x squared plus 4 right parenthesis open parentheses dy over dx close parentheses squared equals straight n squared open parentheses straight y squared plus 4 close parentheses

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397. Differentiate sin^-1(e^x) w.r.t.x.

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398. Differentiate the following w.r.t.x:

sin to the power of negative 1 end exponent left parenthesis straight x square root of straight x right parenthesis comma space 0 less or equal than straight x less or equal than 1

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399. Differentiate the following w.r.t.x:

straight e to the power of sin to the power of negative 1 end exponent straight x end exponent

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400. Differentiate the following w.r.t.x:

straight e to the power of cos minus 1 left parenthesis straight x plus 1 right parenthesis end exponent

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