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Continuity and Differentiability

Short Answer Type

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

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

straight y equals fraction numerator log space straight x over denominator straight x end fraction Differentiating space straight w. straight r. straight t. straight x comma space we space get comma space space space space space dy over dx equals fraction numerator straight x. begin display style straight d over dx end style left parenthesis log space straight x right parenthesis minus log space straight x. begin display style straight d over dx end style left parenthesis straight x right parenthesis over denominator straight x squared end fraction equals fraction numerator straight x. begin display style 1 over straight x end style minus log space straight x.1 over denominator straight x squared end fraction therefore space dy over dx equals fraction numerator 1 minus log space straight x over denominator straight x squared end fraction Differentiating space straight w. straight r. straight t. straight x comma space we space get comma fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator straight x squared begin display style straight d over dx end style left parenthesis 1 minus log space straight x right parenthesis minus left parenthesis 1 minus log space straight x right parenthesis. begin display style straight d over dx end style left parenthesis straight x squared right parenthesis over denominator open parentheses straight x squared close parentheses squared end fraction space space space space space space space space space equals fraction numerator straight x squared begin display style open parentheses negative 1 over straight x close parentheses end style minus left parenthesis 1 minus log space straight x right parenthesis.2 straight x over denominator straight x to the power of 4 end fraction equals fraction numerator negative straight x minus 2 straight x plus 2 straight x space log space straight x over denominator straight x to the power of 4 end fraction space space space space space space space space space equals fraction numerator negative 3 straight x plus 2 xlog space straight x over denominator straight x to the power of 4 end fraction equals fraction numerator negative 3 plus 2 space log space straight x over denominator straight x cubed end fraction fraction numerator straight d cubed straight y over denominator dx cubed end fraction equals fraction numerator straight x cubed open parentheses begin display style 2 over straight x end style close parentheses minus left parenthesis negative 3 plus 2 log space straight x right parenthesis.3 straight x squared over denominator open parentheses straight x cubed close parentheses squared end fraction equals fraction numerator 2 plus 9 minus 6 log space straight x over denominator straight x to the power of 4 end fraction equals fraction numerator 11 minus 6 log space straight x over denominator straight x to the power of 4 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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