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Continuity and Differentiability

Short Answer Type

381. Differentiate x^{sin x}+(sin x)^x w.r.t x.

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382. Differentiate x^{tan x}+(sin x)^{cos x}w.r.t.x.

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383.

If space straight y equals left parenthesis straight x right parenthesis to the power of cos space straight x end exponent plus left parenthesis cos space straight x right parenthesis to the power of sin space straight x end exponent comma space find space dy over dx.

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384.

Find space dy over dx when space left parenthesis cos space straight x right parenthesis to the power of straight y equals left parenthesis cos space straight y right parenthesis to the power of straight x.

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385.

If space straight y equals straight x to the power of cot space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of straight x comma space find space dy over dx.

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386.

If space straight y equals left parenthesis log space straight x right parenthesis to the power of cos space straight x end exponent plus fraction numerator straight x squared plus 1 over denominator straight x squared minus 1 end fraction comma space find space dy over dx

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387. Differentiate the following functions w.r.t. x : $straight x to the power of straight x minus 2 to the power of sin space straight x end exponent$

Let space straight y equals straight x to the power of straight x minus 2 to the power of sin space straight x end exponent Put space straight x to the power of straight x equals straight u comma space 2 to the power of sin space straight x end exponent equals straight v therefore space straight y equals straight u minus straight v therefore space dy over dx equals du over dx minus 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... left parenthesis 1 right parenthesis Now space straight u equals straight x to the power of straight x space rightwards double arrow space log space straight u equals log space straight x to the power of straight x space rightwards double arrow space log space straight u equals straight x space log space straight x therefore space 1 over straight u du over dx equals straight x.1 over straight x plus log space straight x.1 space rightwards double arrow space du over dx equals straight u left parenthesis 1 plus log space straight x right parenthesis therefore space du over dx equals straight x to the power of straight x left parenthesis 1 plus log space straight x right parenthesis 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 Also space straight v equals 2 to the power of sin space straight x end exponent therefore space dv over dx equals 2 to the power of sin space straight x end exponent. log space 2. straight d over dx left parenthesis sin space straight x right parenthesis therefore space dv over dx equals 2 to the power of sin space straight x end exponent. log space 2. cos space straight x 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 3 right parenthesis From space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis comma space we space get comma dy over dx equals straight x to the power of straight x left parenthesis 1 plus log space straight x right parenthesis plus 2 to the power of sin space straight x end exponent. log space 2. space cos space straight x

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

left parenthesis sin space straight x right parenthesis to the power of straight x plus sin to the power of negative 1 end exponent square root of straight x

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389.

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

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

straight x to the power of straight x space cos space straight x end exponent plus fraction numerator straight x squared plus 1 over denominator straight x squared minus 1 end fraction

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