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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.

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

space space space space space space space space space space space straight y equals straight x cubed plus tan space straight x therefore space space dy over dx equals 3 space straight x squared plus sec squared space straight x equals 3 space straight x squared plus 1 plus tan squared space straight x therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals 6 space straight x plus 0 plus 2 space tan space straight x. sec squared straight x therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals 6 space straight x plus 2 space tan space straight x space sec squared space straight x

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530.

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