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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

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389.
Differentiate the following functions w.r.t. x : $straight x to the power of sin space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent$

Let space space space space space space space space straight y equals 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 Put space straight x to the power of sin space straight x end exponent equals straight u comma space left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent equals straight v therefore space straight y equals straight u plus straight v space rightwards double arrow space space dy over dx equals du over dx plus dv over dx straight u equals straight x to the power of sin space straight x end exponent therefore space log space straight u equals log space sin space straight x. log space straight x therefore space 1 over straight u du over dx equals fraction numerator sin space straight x over denominator straight x end fraction plus cos space straight x. log space straight x therefore space du over dx equals straight x to the power of sin space straight x end exponent open square brackets fraction numerator sin space straight x over denominator straight x end fraction plus cos space straight x. log space straight x close square brackets straight v equals left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent therefore space log space straight v equals log left parenthesis sin space straight x right parenthesis to the power of cot space straight x end exponent equals cos space straight x. space log space sin space straight x therefore space 1 over straight v dv over dx equals cos space straight x. fraction numerator 1 over denominator sin space straight x end fraction. cos space straight x minus sin space straight x. log space sin space straight x therefore space dv over dx equals left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent left square bracket cos space straight x. cot space straight x minus sin space straight x. log space sin space straight x right square bracket From space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma we space get comma dy over dx equals straight x to the power of sin space straight x end exponent open square brackets fraction numerator sin space straight x over denominator straight x end fraction plus cos space straight x. log space straight x close square brackets plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent left square bracket cos space straight x. cot space straight x minus sin space straight x. log space sin space straight x right square bracket

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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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Previous Year Papers

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Test Yourself

Short Answer Type

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

389.
Differentiate the following functions w.r.t. x : $straight x to the power of sin space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent$