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Continuity and Differentiability

Short Answer Type

521. Find the second order derivatives of the following functions :x³ log x

Let space space space space straight y equals straight x cubed space log space straight x therefore space dy over dx equals straight x 3.1 over straight x plus log space straight x.3 straight x squared equals equals straight x squared plus 3 straight x squared space log space straight x equals straight x squared left parenthesis 1 plus 3 space log space straight x right parenthesis space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight x squared.3 over straight x plus left parenthesis 1 plus 3 space log space straight x right parenthesis.2 straight x space space space space space space space space space space space space space space equals 3 straight x plus 2 straight x left parenthesis 1 plus 3 space log space straight x right parenthesis equals straight x left square bracket 3 plus 2 left parenthesis 1 plus 3 space log space straight x right parenthesis right square bracket space space space space space space space space space space space space space space equals straight x left square bracket 3 plus 2 plus 6 space log space straight x right square bracket equals straight x left parenthesis 5 plus 6 space log space straight x right parenthesis

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522. Find the second order derivatives of the following functions : e^x sin 5x

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523. Find the second order derivatives of the following functions : e^{6 x} cos 3x

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524. Find the second order derivatives of the following functions : tan^-1 x

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525. Find the second order derivatives of the following functions : log (log x)

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526. Find the second order derivatives of the following functions : sin (log x)

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527.

If space straight x equals square root of straight a to the power of sin to the power of negative 1 end exponent straight t end exponent end root comma space straight y equals square root of straight a to the power of cos to the power of negative 1 end exponent straight t end exponent end root comma space show space that dy over dx equals negative straight y over straight x.

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528.

Find space fraction numerator straight d cubed straight y over denominator dx cubed end fraction space when space straight y equals fraction numerator log space straight x over denominator straight x end fraction.

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529.

Find space fraction numerator straight d squared straight y over denominator dx squared end fraction comma space if space space straight y equals straight x cubed plus tan space straight x

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530.

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