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

left parenthesis straight x space cos space straight x right parenthesis to the power of straight x plus left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent

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

straight e to the power of sin space straight x end exponent plus left parenthesis tan space straight x right parenthesis to the power of straight x

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

left parenthesis sin space straight x right parenthesis to the power of tan space space straight x end exponent plus left parenthesis cos space straight x right parenthesis to the power of sec space straight x end exponent

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

straight x to the power of tan 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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395. Differentiate the following functions w.r.t. x :

straight x to the power of straight x to the power of straight x end exponent plus straight x to the power of sin space straight x end exponent

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396. If x=sec ө-cos ө, y=secⁿө-cosⁿ ө then show that

left parenthesis straight x squared plus 4 right parenthesis open parentheses dy over dx close parentheses squared equals straight n squared open parentheses straight y squared plus 4 close parentheses

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397. Differentiate sin^-1(e^x) w.r.t.x.

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Let space space space space space straight y equals sin to the power of negative 1 end exponent left parenthesis straight x square root of straight x right parenthesis equals sin to the power of negative 1 end exponent open parentheses straight x to the power of 3 over 2 end exponent close parentheses therefore space dy over dx equals fraction numerator 1 over denominator square root of 1 minus open parentheses straight x to the power of 3 over 2 end exponent close parentheses squared end root end fraction. straight d over dx open parentheses straight x to the power of 3 over 2 end exponent close parentheses equals fraction numerator 1 over denominator square root of 1 minus straight x cubed end root end fraction.3 over 2 straight x to the power of 1 half end exponent therefore space dy over dx equals fraction numerator 3 square root of straight x over denominator 2 square root of 1 minus straight x cubed end root end fraction

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399. Differentiate the following w.r.t.x:

straight e to the power of sin to the power of negative 1 end exponent straight x end exponent

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400. Differentiate the following w.r.t.x:

straight e to the power of cos minus 1 left parenthesis straight x plus 1 right parenthesis end exponent

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