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Continuity And Differentiability

Question
CBSEENMA12035391
Wired Faculty App

If space straight y space equals straight A space straight e to the power of straight m space straight x end exponent plus straight B space straight e to the power of straight n space straight x end exponent comma space show space that space dy over dx minus left parenthesis straight m plus straight n right parenthesis dy over dx plus straight m space straight n space straight y equals 0

Solution
straight y space equals straight A space straight e to the power of straight m space straight x end exponent plus straight B space straight e to the power of straight n space straight x end exponent 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 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 therefore space dy over dx equals straight A space straight m space straight e to the power of straight m space straight x end exponent plus straight B space straight n space straight e to the power of straight n space straight x end exponent 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 space space space space space space space space... left parenthesis 2 right parenthesis and space fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight A space straight m squared space straight e to the power of straight m space straight x end exponent plus straight B space straight n squared space straight e to the power of straight n space straight x end exponent 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 3 right parenthesis straight L. straight H. straight S. space equals fraction numerator straight d squared straight y over denominator dx squared end fraction minus left parenthesis straight m plus straight n right parenthesis dy over dx plus straight m space straight n space straight y space space space space space space space space space space space space space equals left parenthesis straight A space straight m squared space straight e to the power of straight m space straight x end exponent plus straight B space straight n squared space straight e to the power of straight n space straight x end exponent right parenthesis minus left parenthesis straight m plus straight n right parenthesis space left parenthesis straight A space straight m space straight e to the power of straight m space straight x end exponent plus straight B space straight n space straight e to the power of straight n space straight x end exponent right parenthesis 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 plus straight m space straight n left parenthesis straight A space straight m space straight e to the power of straight m space straight x end exponent plus straight B space straight n space straight e to the power of straight n space straight x end exponent right parenthesis 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 comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis right square bracket space space space space space space space space space space space space space equals straight A space straight m squared space straight e to the power of straight m space straight x end exponent plus straight B space straight n squared space straight e to the power of straight n space straight x end exponent minus straight A space straight m squared space straight e to the power of straight m space straight x end exponent minus straight B space straight m space straight n space straight e to the power of straight n space straight x end exponent minus straight A space straight m space straight n space straight e to the power of straight m space straight x end exponent 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 minus straight B space straight n squared space straight e to the power of straight n space straight x end exponent plus straight A space straight m space straight n space straight e to the power of straight m space straight x end exponent plus straight B space straight m space straight n space straight e to the power of straight n space straight x end exponent space space space space space space space space space space space space space equals 0 space space space space space space space space space space space space space equals straight R. straight H. straight S.

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