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Continuity and Differentiability

Short Answer Type

391. Differentiate the following functions w.r.t. x : $left parenthesis straight x space cos space straight x right parenthesis to the power of straight x plus left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent$

Let space space space space straight y equals 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 Put space left parenthesis straight x space cos space straight x right parenthesis to the power of straight x equals straight u comma space left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over 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... left parenthesis 1 right parenthesis Now space straight u equals left parenthesis straight x space cos space straight x right parenthesis to the power of straight x space space space space space space rightwards double arrow space log space straight u equals log left parenthesis straight x space cos space straight x right parenthesis to the power of straight x therefore space log space straight u equals straight x. log left parenthesis straight x space cos space straight x right parenthesis therefore 1 over straight u du over dx equals straight x. fraction numerator straight x left parenthesis negative sin space straight x right parenthesis plus cos space straight x.1 over denominator straight x space cos space straight x end fraction plus log left parenthesis straight x space cos space straight x right parenthesis.1 therefore space du over dx equals left parenthesis straight x space cos space straight x right parenthesis to the power of straight x open square brackets fraction numerator cos space straight x minus straight x space sin space straight x over denominator cos space straight x end fraction plus log left parenthesis straight x space cos space straight x right parenthesis close square brackets 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 Also space straight v equals left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent space space space space rightwards double arrow space log space straight v equals log left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent therefore space space space space space space log space straight v equals 1 over straight x. log left parenthesis straight x space sin space straight x right parenthesis therefore space 1 over straight v dv over dx equals 1 over straight x. fraction numerator straight x space cos space straight x plus sin space straight x.1 over denominator straight x space sin space straight x end fraction plus log left parenthesis straight x space sin space straight x right parenthesis. open parentheses negative 1 over straight x squared close parentheses space space space space space space space space space space space space space space space space space space space equals fraction numerator straight x space cos space straight x plus sin space straight x over denominator straight x squared space sin space straight x end fraction minus fraction numerator log left parenthesis straight x space sin space straight x right parenthesis over denominator straight x squared end fraction therefore 1 over straight v dv over dx equals fraction numerator straight x space cos space straight x plus sin space straight x minus sin space straight x. log left parenthesis straight x space sin space straight x right parenthesis over denominator straight x squared space sin space straight x end fraction therefore space dv over dx equals left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent open square brackets fraction numerator straight x space cos space straight x plus sin space straight x minus sin space straight x. log left parenthesis straight x space sin space straight x right parenthesis over denominator straight x squared space sin space straight x end fraction close square brackets 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 space we space get comma dy over dx equals left parenthesis straight x space cos space straight x right parenthesis to the power of straight x open square brackets fraction numerator cos space straight x minus straight x space sin space straight x over denominator cos space straight x end fraction plus log left parenthesis straight x space cos space straight x right parenthesis close square brackets plus left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent open square brackets fraction numerator straight x space cos space straight x plus sin space straight x minus sin space straight x. log left parenthesis straight x space sin space straight x right parenthesis over denominator straight x squared space sin space straight x end fraction close square brackets

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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

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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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