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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

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

space space space space space space space space space space straight y space equals space cos to the power of negative 1 end exponent space straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis therefore space dy over dx equals negative fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction equals negative open parentheses 1 minus straight x squared close parentheses to the power of negative 1 half end exponent therefore space dy over dx equals 1 half open parentheses 1 minus straight x squared close parentheses to the power of 3 over 2 end exponent left parenthesis negative 2 straight x right parenthesis equals negative straight x over open parentheses 1 minus straight x squared close parentheses to the power of 3 over 2 end exponent equals negative fraction numerator cos space straight y over denominator open parentheses 1 minus cos squared close parentheses to the power of 3 over 2 end exponent end fraction space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket space space space space space space space space space space space space space equals negative fraction numerator cos space straight y over denominator open parentheses 1 minus cos squared close parentheses to the power of 3 over 2 end exponent end fraction equals negative fraction numerator cos space straight y over denominator sin cubed space straight y end fraction equals negative fraction numerator cos space straight y over denominator sin space straight y end fraction. fraction numerator 1 over denominator sin squared space straight y end fraction equals negative cot space straight y space cosec squared space straight y.

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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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