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Continuity And Differentiability

Question
CBSEENMA12035440
Wired Faculty App

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.

Solution
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

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