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Continuity and Differentiability

Short Answer Type

371. $Differentiate space cos space straight x times cos space 2 straight x. cos 3 straight x space space straight w. straight r. straight t. straight x.$

Let space space space space space space straight y space equals space cos space straight x space times space cos space 2 straight x space cos space 3 space straight x therefore space log space straight y space equals space log space left square bracket cos space straight x space times space cos space 2 straight x space times space cos space 3 straight a right square bracket therefore space log space straight y space equals space log space cos space straight x space plus space log space cos space 2 straight a space plus space log space cos space 3 straight x Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get. space 1 over straight y dy over dx equals fraction numerator negative sin space straight x over denominator cos space straight x end fraction plus fraction numerator negative 2 sin space straight x over denominator cos space 2 straight x end fraction plus fraction numerator negative 3 sin space 3 straight x over denominator cos space 3 straight x end fraction therefore space space space dy over dx equals negative straight y left square bracket tan space straight x plus 2 tan space 2 straight x plus 3 space tan space 3 straight x right square bracket therefore space space space space space space space space space space space equals negative cos space straight x space cos 2 straight x space cos space 3 straight x left square bracket tan space straight x plus 2 tan space 2 straight x plus 3 tan space 3 straight x right square bracket

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372.

Find space dy over dx comma space if space straight y equals left parenthesis sin space straight x right parenthesis to the power of sin space straight x to the power of sin space straight x... infinity end exponent end exponent

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373.

If space straight y equals left parenthesis cos right parenthesis to the power of cos space straight x to the power of cos space straight x..... infinity end exponent end exponent comma space prove space that space dy over dx equals negative fraction numerator straight y squared tan space straight x over denominator 1 minus straight y space log space cos space straight x end fraction

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374.

If space straight y equals left parenthesis tan space straight x right parenthesis to the power of tan space straight x to the power of tan space straight x.... infinity end exponent end exponent comma space prove space that space dy over dx equals 2 space at space straight x equals straight pi over 2

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375. Differentiate (log x)^{cos x} w.r.t. x.

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376. Differentiate (log x)^{sin x} w.r.t.x.

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377. Differentiate x^{sin x}w.r.t. (sin x)^x

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378. Differentiate sin x²w.r.t.x².

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379. Differentiate cos(x^x) w.r.t.x.

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380.

Differentiate space straight x to the power of tan space straight x end exponent plus square root of fraction numerator straight x squared plus 1 over denominator straight x end fraction end root straight w. straight r. straight t. straight x.

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