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

Here space space space space space space 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 Put space left parenthesis log space straight x right parenthesis to the power of cos space straight x end exponent equals straight u comma space fraction numerator straight x squared plus 1 over denominator straight x squared minus 1 end fraction equals straight v 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 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 therefore space log space straight u equals log open square brackets left parenthesis log space straight x right parenthesis to the power of cos space straight x end exponent close square brackets therefore space log space straight u equals cos space straight x. space loh left parenthesis log space straight x right parenthesis therefore space 1 over straight u du over dx equals cos space straight x. open parentheses fraction numerator 1 over denominator log space straight x end fraction.1 over straight x close parentheses plus log left parenthesis log space straight x right parenthesis. left parenthesis negative sin space straight x right parenthesis therefore space du over dx equals straight u open square brackets fraction numerator cos space straight x over denominator straight x space log space straight x end fraction minus sin space straight x space log left parenthesis log space straight x right parenthesis close square brackets therefore space du over dx equals left parenthesis log space straight x right parenthesis to the power of cos space straight x end exponent open square brackets fraction numerator cos space straight x over denominator straight x space log space straight x end fraction minus sin space straight x space log left parenthesis log 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... left parenthesis 2 right parenthesis

Again space straight v equals fraction numerator straight x squared plus 1 over denominator straight x squared minus 1 end fraction therefore space dv over dx equals fraction numerator left parenthesis straight x squared minus 1 right parenthesis. begin display style straight d over dx end style left parenthesis straight x squared plus 1 right parenthesis minus left parenthesis straight x squared plus 1 right parenthesis begin display style straight d over dx end style left parenthesis straight x squared minus 1 right parenthesis over denominator left parenthesis straight x squared minus 1 right parenthesis squared end fraction therefore space dv over dx equals negative fraction numerator 4 straight x over denominator left parenthesis straight x squared minus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 space space 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 log space straight x right parenthesis to the power of cos space straight x end exponent open square brackets fraction numerator cos space straight x over denominator straight x space log space straight x end fraction minus sin space straight x. log left parenthesis log space straight x right parenthesis close square brackets minus fraction numerator 4 straight x over denominator left parenthesis straight x squared minus 1 right parenthesis squared end fraction

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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

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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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