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Continuity And Differentiability

Question
CBSEENMA12035425
Wired Faculty App

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.

Solution
Here space space space 2 space straight y equals straight x open parentheses 1 plus dy over dx close parentheses Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma space space space space space space space space 2 dy over dx equals straight x open parentheses 0 plus fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses plus open parentheses 1 plus dy over dx close parentheses plus 1 or space space space space space 2 dy over dx equals straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus 1 plus dy over dx or space space space space space space space dy over dx equals straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus 1 Again space differentiating space both space sides space straight w. straight r. straight t. straight x comma space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight x fraction numerator straight d cubed straight y over denominator dx cubed end fraction plus fraction numerator straight d squared straight y over denominator dx squared end fraction.1 plus 0 rightwards double arrow space straight x fraction numerator straight d cubed straight y over denominator dx cubed end fraction equals 0 space space space space space space space space space space space space rightwards double arrow space fraction numerator straight d cubed straight y over denominator dx cubed end fraction equals 0 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 straight x not equal to 0 right square bracket rightwards double arrow fraction numerator straight d over denominator space dx end fraction open parentheses fraction numerator straight d cubed straight y over denominator dx cubed end fraction close parentheses equals 0 space space rightwards double arrow fraction numerator straight d squared straight y over denominator dx squared end fraction equals space constant. space space space space space space open square brackets because space straight d over dx left parenthesis straight c right parenthesis equals 0 close square brackets

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