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

Let space space space space space space space space straight y equals 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 Put space straight e to the power of sin space straight x end exponent equals straight u comma space left parenthesis tan space straight x right parenthesis to the power of straight x equals straight u therefore space space space space space space space straight y equals straight u plus straight v therefore space dy over dx equals du over dx plus dv over dx Now space straight u equals straight e to the power of sin space straight x end exponent therefore space du over dx equals straight e to the power of sin space straight x end exponent. cos space straight x Also space space space space space space space straight v equals left parenthesis tan space straight x right parenthesis to the power of straight x therefore space space space space space log space straight v equals log left parenthesis tan space straight x right parenthesis to the power of straight x therefore space space space space space log space straight v equals straight x. log left parenthesis tan space straight x right parenthesis therefore space 1 over straight v dv over dx equals straight x. fraction numerator sec squared straight x over denominator tan space straight x end fraction plus log space tan space straight x.1 therefore space space space space space space space dv over dx equals straight v open square brackets straight x. fraction numerator 1 over denominator cos squared straight x end fraction. fraction numerator cos space straight x over denominator sin space straight x end fraction plus log space tan space straight x close square brackets therefore space space space space space space space dv over dx equals left parenthesis tan space straight x right parenthesis to the power of straight x open square brackets fraction numerator straight x over denominator sin space xcos space straight x end fraction plus log space tan space straight x close square brackets 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 space space space space space space space space space space space dy over dx equals straight e to the power of sin space straight x end exponent. cos space straight x plus left parenthesis tan space straight x right parenthesis to the power of straight x open square brackets fraction numerator straight x over denominator sin space xcos space straight x end fraction plus log space tan space straight x close square brackets

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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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