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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.

81 Views

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.

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

We space have 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 Differentiating space both space sides space of space left parenthesis 1 right parenthesis space straight w. straight r. straight t comma space we space get comma 1 half straight x to the power of negative 1 half end exponent plus 1 half straight y to the power of negative 1 half end exponent equals 0 space space space space space space space space or space dy over dx equals negative straight x to the power of negative 1 half end exponent over straight y to the power of negative 1 half end exponent equals negative straight y to the power of 1 half end exponent over straight x to the power of 1 half end exponent space space space space space space space space space... left parenthesis 2 right parenthesis Differentiating space again comma space we space get comma space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative fraction numerator straight x to the power of 1 half end exponent. begin display style 1 half end style straight y to the power of negative 1 half end exponent. begin display style dy over dx end style minus straight y to the power of negative 1 half end exponent. begin display style 1 half end style straight x to the power of negative 1 half end exponent over denominator open parentheses straight x to the power of 1 half end exponent close parentheses squared end fraction space space space space space space space space space space space space space space space space equals negative 1 half. fraction numerator begin display style straight x to the power of 1 half end exponent over straight y to the power of 1 half end exponent end style open square brackets negative begin display style straight y to the power of 1 half end exponent over straight x to the power of 1 half end exponent end style close square brackets minus begin display style straight y to the power of 1 half end exponent over straight x to the power of 1 half end exponent end style over denominator straight x end fraction space space space space space space space space open square brackets Putting space the space value space of space dy over dx from space left parenthesis 2 right parenthesis close square brackets space space space space space space space space space space space space space space space space equals 1 half fraction numerator 1 plus begin display style straight y to the power of 1 half end exponent over straight x to the power of 1 half end exponent end style over denominator straight x end fraction equals 1 half. fraction numerator straight x to the power of 1 half end exponent plus straight y to the power of 1 half end exponent over denominator straight x to the power of 3 over 2 end exponent end fraction equals 1 half. straight a to the power of 1 half end exponent over straight x to the power of 3 over 2 end exponent 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 Putting space straight x equals straight a comma space the space value space of space equals fraction numerator straight d squared straight y over denominator dx squared end fraction equals 1 half straight a to the power of 1 half end exponent over straight a to the power of 3 over 2 end exponent equals fraction numerator 1 over denominator 2 space straight a end fraction

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