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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.

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

space space space space space space straight y equals left parenthesis sin space straight x right parenthesis sin space xsin space straight x space... space infinity space therefore space straight y space equals space left parenthesis sin space straight x right parenthesis straight y therefore space log space straight y space equals space log space left parenthesis sin right parenthesis straight y space space rightwards double arrow space log space straight y space equals straight y space times space log space left parenthesis sin space straight x right parenthesis space space space space... left parenthesis 1 right parenthesis space space Differentiating space straight w. straight r. straight t. straight x comma space we space get comma space 1 over straight y dy over dx equals straight y. fraction numerator cos space straight x over denominator sin space straight x end fraction plus log left parenthesis sin space straight x right parenthesis. dy over dx therefore space open square brackets 1 over straight y minus log left parenthesis sin space straight x right parenthesis close square brackets dy over dx equals straight y space cot space straight x therefore space open parentheses fraction numerator 1 minus straight y space log left parenthesis sin space straight x right parenthesis over denominator straight y end fraction close parentheses dy over dx equals straight y space cot space straight x therefore space dy over dx equals fraction numerator straight y squared cot space straight x over denominator 1 minus ylog left parenthesis sin space straight x right parenthesis end fraction therefore space dy over dx equals fraction numerator straight y squared cot space straight x over denominator 1 minus log space straight y 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 left square bracket because space of left parenthesis 1 right parenthesis right square bracket

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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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