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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

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388. Differentiate the following functions w.r.t. x : $left parenthesis sin space straight x right parenthesis to the power of straight x plus sin to the power of negative 1 end exponent square root of straight x$

Let space straight y equals 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 Put space left parenthesis sin space straight x right parenthesis to the power of straight x equals straight u comma space sin to the power of negative 1 end exponent square root of straight x 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... left parenthesis 1 right parenthesis Now space straight u equals left parenthesis sin space straight x right parenthesis to the power of straight x space rightwards double arrow space log space straight u equals log left parenthesis sin space straight x right parenthesis to the power of straight x therefore space log space straight u equals straight x. space log left parenthesis sin space straight x right parenthesis to the power of straight x therefore space 1 over straight u du over dx equals straight x. fraction numerator cos space straight x over denominator sin space straight x end fraction plus log left parenthesis sin space straight x right parenthesis.1 therefore space du over dx equals left parenthesis sin space straight x right parenthesis to the power of straight x left square bracket straight x space cot space straight x plus log left parenthesis sin space straight x right parenthesis right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 sin to the power of negative 1 end exponent square root of straight x therefore space dv over dx equals fraction numerator 1 over denominator square root of 1 minus left parenthesis square root of straight x right parenthesis squared end root end fraction. straight d over dx left parenthesis square root of straight x right parenthesis equals fraction numerator 1 over denominator square root of 1 minus straight x end root end fraction. fraction numerator 1 over denominator 2 square root of straight x end fraction therefore space dv over dx equals fraction numerator 1 over denominator 2 square root of straight x square root of 1 minus straight x end root 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 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 sin space straight x right parenthesis to the power of straight x left square bracket straight x space cot space straight x plus log left parenthesis sin space straight x right parenthesis right square bracket plus fraction numerator 1 over denominator 2 square root of straight x square root of 1 minus straight x end root end fraction

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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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