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Continuity And Differentiability

Question
CBSEENMA12035414
Wired Faculty App

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.

Solution
Here space space 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 therefore space space space space space straight y over straight x equals log space straight x minus log left parenthesis straight a plus straight b space straight x right parenthesis Differentaiting space straight w. straight r. tx comma space we space get comma space space space space space fraction numerator straight x begin display style dy over dx end style minus straight y.1 over denominator straight x squared end fraction equals 1 over straight x minus fraction numerator straight b over denominator straight a plus straight b space straight x end fraction space space space space space space space straight x dy over dx minus straight y equals straight x minus straight b space straight x squared straight a plus straight b space straight x therefore space space space straight x dy over dx minus straight y equals fraction numerator straight a space straight x plus straight b space straight x squared minus straight b space straight x squared over denominator straight a plus straight b space straight x end fraction therefore space space space straight x dy over dx minus straight y equals fraction numerator straight a space straight x over denominator straight a plus straight b space straight x end fraction 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 Again space differentiating space straight w. straight r. straight t. straight x comma space we space get comma space space space space space space straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx.1 minus dy over dx equals fraction numerator left parenthesis straight a plus straight b space straight x right parenthesis. straight a space minus straight a space straight x. space straight b over denominator left parenthesis straight a plus straight b space straight x right parenthesis squared end fraction therefore space space space space straight x fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator straight a squared over denominator left parenthesis straight a plus straight b space straight x right parenthesis squared end fraction rightwards double arrow space space straight x cubed fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator straight a squared straight x squared over denominator left parenthesis straight a plus straight b space straight x right parenthesis squared end fraction rightwards double arrow space 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 close parentheses 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 left square bracket because space of space left parenthesis 1 right parenthesis right square bracket

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