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Continuity and Differentiability

Short Answer Type

531.

If space straight x equals log space straight theta comma space straight theta greater than 0 space and space straight y equals 1 over straight theta space then space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx equals 0.

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532. $If space space straight y equals straight e to the power of straight x left parenthesis sin space straight x plus cos space straight x right parenthesis comma space prove space that space dy over dx minus 2 dy over dx plus 2 straight y equals 0$

space space space space space space space space space space space space straight y equals straight e to the power of straight x left parenthesis sin space straight x plus cos space straight x right parenthesis therefore space space dy over dx equals straight e to the power of straight x left parenthesis cos space straight x minus sin space straight x right parenthesis plus left parenthesis sin space straight x plus cos space straight x right parenthesis. straight e to the power of straight x space space space space space space space space space space space space space space equals straight e to the power of straight x left parenthesis cos space straight x minus sin space straight x plus sin space straight x plus cos space straight x right parenthesis therefore space space dy over dx equals 2 straight e to the power of straight x space cos space straight x therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals 2 left square bracket straight e to the power of straight x left parenthesis negative sin space straight x right parenthesis plus cos space straight x. space straight e to the power of straight x right square bracket therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals 2 straight e to the power of straight x left parenthesis cos space straight x minus sin space straight x right parenthesis straight L. straight H. straight S equals fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 dy over dx plus 2 straight y space space space space space space space space space space space equals 2 e to the power of x left parenthesis cos space straight x minus sin space straight x right parenthesis minus 2 left parenthesis 2 e to the power of x space cos space straight x right parenthesis plus 2 e to the power of x left parenthesis sin space straight x plus cos space straight x right parenthesis space space space space space space space space space space space equals 2 e to the power of x space cos space straight x minus 2 space e to the power of x space sin space straight x minus 4 space e to the power of x space cos space straight x plus 2 space e to the power of x space sin space straight x plus 2 space e to the power of x space cos space straight x space space space space space space space space space space space equals 0 equals straight R. straight H. straight S.

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533.

If space straight y equals 5 space cos space straight x minus 3 space sin space straight x comma space prove space that space fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0.

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534.

Find space fraction numerator straight d squared straight y over denominator dx squared end fraction comma space if space straight y equals straight e to the power of straight x space sin end exponent space to the power of straight x.

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535.

If space straight y space equals space cos to the power of negative 1 end exponent space straight x comma space find space fraction numerator straight d squared straight y over denominator dx squared end fraction space in space terms space of space straight y space alone.

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536. If y = sin (sin x), prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction plus tan space straight x space dy over dx plus space straight y space cos squared space straight x equals 0

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537. If y = sin (log x), prove that

straight x 2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight y equals 0.

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538. f y = a sin 2x + b cos 2x, show that

fraction numerator d squared y over denominator d x squared end fraction plus 4 space y equals 0.

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539. If y=A sin x+B cos x, prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0

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540. If y = A cos n x + B sin n x, prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight n squared space straight y equals 0

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