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Continuity and Differentiability

Short Answer Type

561.

If space straight y equals log open parentheses straight x plus square root of 1 plus straight x squared end root 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 straight x dy over dx equals 0.

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562. $If space straight y equals fraction numerator sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction comma space prove space that space left parenthesis 1 minus straight x squared right parenthesis straight y subscript 2 minus 3 space straight x space straight y subscript 1 minus straight y equals 0.$

straight y equals fraction numerator sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction space space space space space rightwards double arrow space square root of 1 minus straight x squared end root space straight y equals sin to the power of negative 1 end exponent straight x Diff. space straight w. straight r. straight t. straight x. space space straight y subscript 1 plus straight y. fraction numerator negative 2 space straight x over denominator 2 square root of 1 minus straight x squared end root end fraction equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction or space space space square root of 1 minus straight x squared end root space straight y subscript 1 minus fraction numerator straight x space straight y over denominator square root of 1 minus straight x squared end root end fraction equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction or space space space space left parenthesis 1 minus straight x squared right parenthesis straight y subscript 1 minus straight x space straight y equals 1. again space differentiating space straight w. straight r. straight t. straight x comma space left parenthesis 1 minus straight x squared right parenthesis. straight y subscript 2 plus straight y subscript 1 left parenthesis negative 2 straight x right parenthesis minus space xy subscript 1 minus straight y subscript 1 equals 0 or space left parenthesis 1 minus straight x squared right parenthesis. straight y subscript 2 minus 3 space straight x space minus straight y equals 0

82 Views

563.

If space space space straight x equals straight a left parenthesis straight ө plus sin space straight ө right parenthesis comma space straight y equals straight a left parenthesis 1 minus cos space straight ө right parenthesis comma space find space dy over dx space at space straight theta equals straight pi over 2.

89 Views

564.

If space space straight x equals straight a left parenthesis cos space straight t plus straight t space sin space straight t right parenthesis comma space straight y equals straight a left parenthesis sin space straight t space minus straight t space cos space straight t right parenthesis comma space evaluate space fraction numerator straight d squared straight y over denominator dx squared end fraction space at space straight t equals straight pi over 4

79 Views

565.

If space straight x equals 3 space sin space straight t minus sin space 3 straight t comma space straight y equals 3 space cos space straight t minus cos space 3 space straight t comma space find space dy over dx space at space straight t equals straight pi over 3.

87 Views

566.

If space space space straight x equals 2 space cos space straight ө minus cos space 2 straight ө space and space straight y equals 2 space sin space straight ө minus sin space 2 straight ө comma space find space open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses subscript straight theta equals straight pi over 2 end subscript

81 Views

567.

If space straight x to the power of 1 half end exponent plus straight y to the power of 1 half end exponent equals straight a to the power of 1 half end exponent comma space find space the space value space of space fraction numerator straight d squared straight y over denominator dx squared end fraction space at space straight x equals straight a.

79 Views

568.

If space straight x squared over straight a squared plus straight y squared over straight b squared equals 1 comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative fraction numerator straight b 4 over denominator straight a squared straight y cubed end fraction.

82 Views

569.

If space straight y equals tan to the power of negative 1 end exponent space straight x comma space show space that space open parentheses 1 plus straight x squared close parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction plus 2 space straight x dy over dx equals 0

93 Views

570.

If space straight y equals left parenthesis tan to the power of negative 1 end exponent space straight x right parenthesis squared comma space show space that space left parenthesis straight x squared space plus space 1 right parenthesis 2 space straight y subscript 2 plus 2 space straight x left parenthesis straight x squared plus 1 right parenthesis space straight y to the power of 1 equals 2.

97 Views