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Continuity and Differentiability

Short Answer Type

551. If y=e^{a x}sin bx, prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight a dy over dx plus left parenthesis straight a squared plus straight b squared right parenthesis straight y equals 0

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552.

If space straight y equals straight x space log open parentheses fraction numerator straight x over denominator straight a plus straight b space straight x end fraction close parentheses comma space prove space that space straight x cubed fraction numerator straight d squared straight y over denominator dx squared end fraction equals open parentheses straight x dy over dx minus straight y squared close parentheses.

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553.

If space straight x equals tan open parentheses 1 over straight a log space straight y close parentheses comma space show space that space left parenthesis 1 plus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus left parenthesis 2 space straight x minus straight a right parenthesis dy over dx equals 0.

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554. If y=x^x, prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction minus 1 over straight y open parentheses dy over dx close parentheses squared minus straight y over straight x equals 0.

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555.

If space space space space straight y equals left parenthesis straight a plus straight b space straight t right parenthesis space straight e to the power of straight n space straight t end exponent comma space prove space that space fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight n dy over dx plus straight n squared straight y equals 0.

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556.

If space straight y equals fraction numerator straight a space straight x plus straight b over denominator straight c space straight x plus straight d end fraction comma space prove space that space 2 straight y subscript 1 space straight y subscript 3 equals 3 space straight y subscript 2 squared.

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557.

If space straight y equals log space left parenthesis 1 plus cos space straight x right parenthesis comma space prove space that space straight y subscript 1 space straight y subscript 2 plus straight y subscript 3 equals 0.

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558.

If space 2 space straight y equals straight x open parentheses 1 plus dy over dx close parentheses comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction space is space constant space.

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559. If y = (tan^-1 x)², then prove that (1 + x²)² y₂ + 2x(1 + x²)y_{1 = 2.}

space space space space space space space space straight y equals left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis squared 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 space space straight y subscript 1 equals 2 space tan to the power of negative 1 end exponent straight x. fraction numerator 1 over denominator 1 plus straight x squared end fraction rightwards double arrow space left parenthesis 1 plus straight x squared right parenthesis straight y subscript 1 equals 2 space tan to the power of negative 1 end exponent straight x rightwards double arrow space space left parenthesis 1 plus straight x squared right parenthesis squared straight y subscript 1 squared equals 4 open parentheses tan to the power of negative 1 end exponent straight x close parentheses squared rightwards double arrow space space left parenthesis 1 plus straight x squared right parenthesis squared straight y subscript 1 squared equals 4 straight y 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 square bracket because space of space left parenthesis 1 right parenthesis right square bracket Differentiation space both space sides space straight w. straight r. straight t. straight x comma space space space space space space space left parenthesis 1 plus straight x squared right parenthesis squared.2 straight y subscript 1 straight y subscript 2 plus 2 left parenthesis 1 plus straight x squared right parenthesis.2 straight x. straight y subscript 1 squared equals 4 straight y subscript 1 Divides space both space sides space ny space 2 straight y subscript 1 comma space we space get comma left parenthesis 1 plus straight x squared right parenthesis squared straight y subscript 2 plus 2 straight x left parenthesis 1 plus straight x squared right parenthesis straight y subscript 1 equals 2

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560.

If space straight y equals sin to the power of negative 1 end exponent straight x comma space then space show space that space open parentheses 1 minus straight x squared close parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x dy over dx equals 0.

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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

559. If y = (tan-1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2.

559. If y = (tan^-1 x)², then prove that (1 + x²)² y₂ + 2x(1 + x²)y_{1 = 2.}