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

Here space space space 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 therefore space space space log left parenthesis cos space straight x right parenthesis to the power of straight y equals log left parenthesis cos space straight y right parenthesis to the power of straight x therefore space straight y. space log left parenthesis cos space straight x right parenthesis equals straight x. space log left parenthesis cos space straight y right parenthesis Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma space space straight y. fraction numerator negative sin space straight x over denominator cos space straight x end fraction plus log left parenthesis cos space straight x right parenthesis. dy over dx equals straight x. fraction numerator negative sin space straight y over denominator cos space straight y end fraction dy over dx plus log left parenthesis cos space straight y right parenthesis.1 therefore space minus straight y space tan space straight x plus log left parenthesis cos space straight x right parenthesis. dy over dx equals negative straight x space tan space straight y dy over dx plus log left parenthesis cos space straight y right parenthesis therefore space left square bracket log left parenthesis cos space straight x right parenthesis plus straight x space tan space straight y right square bracket dy over dx equals straight y space tan space straight x plus log left parenthesis cos space straight y right parenthesis therefore space dy over dx equals fraction numerator straight y space tan space straight x plus log left parenthesis cos space straight y right parenthesis over denominator log left parenthesis cos space straight x right parenthesis plus straight x space tan space straight y end fraction

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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

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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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