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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

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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

Let space space space space space space straight y equals straight x to the power of sin space straight x end exponent comma space straight u equals left parenthesis sin space straight x right parenthesis to the power of straight x therefore space log space straight y equals log space straight x to the power of sin space straight x end exponent equals sin space straight x. log space straight x space rightwards double arrow space 1 over straight y dy over dx equals sin space straight x.1 over straight x plus log space straight x. cos space straight x therefore space space dy over dx equals space 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 Also space space space straight u equals left parenthesis sin space straight x right parenthesis to the power of straight x therefore space log space straight u equals log left parenthesis sin space straight x right parenthesis to the power of straight x equals straight x. log left parenthesis sin space straight x right parenthesis therefore space 1 over straight u du over dx equals straight x. straight d over dx left square bracket log left parenthesis sin space straight x right parenthesis right square bracket plus log left parenthesis sin space straight x right parenthesis. straight d over dx left parenthesis straight x right parenthesis therefore space du over dx equals straight u open square brackets straight x.1 over sinx. cos space straight x plus log left parenthesis sin space straight x right parenthesis.1 close square brackets 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 Now space dy over du equals fraction numerator begin display style dy over dx end style over denominator begin display style du over dx end style end fraction equals fraction numerator straight x to the power of sin space straight x end exponent open square brackets begin display style fraction numerator sin space straight x over denominator straight x end fraction end style plus cos space straight x. log space straight x close square brackets over denominator 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 end fraction

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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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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

377. Differentiate xsin xw.r.t. (sin x)x

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