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Continuity And Differentiability

Question
CBSEENMA12034808
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If space straight e to the power of straight x plus straight e to the power of straight y equals straight e to the power of straight x plus straight y comma end exponent prove space that space dy over dx equals negative fraction numerator straight e to the power of straight x left parenthesis straight e to the power of straight y minus 1 right parenthesis over denominator straight e to the power of straight y left parenthesis straight e to the power of straight y minus 1 right parenthesis end fraction

Solution
Here space straight e to the power of straight x plus straight e to the power of straight y equals straight e to the power of straight x plus straight y end exponent Differentiating space both space sides space straight w. straight r. straight t. space straight x comma space we space get comma bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold e to the power of bold x bold plus bold e to the power of bold y bold dy over bold dx bold equals bold e to the power of bold x bold plus bold y end exponent open parentheses bold 1 bold plus bold dy over bold dx close parentheses bold therefore bold space bold space bold space bold space bold space bold space bold space bold e to the power of bold x bold plus bold e to the power of bold y bold dy over bold dx bold equals bold e to the power of bold x bold plus bold y end exponent bold plus bold e to the power of bold x bold plus bold y end exponent bold dy to the power of bold 1 over bold dx bold therefore bold space bold left parenthesis bold e to the power of bold x bold minus bold e to the power of bold x bold plus bold y end exponent bold right parenthesis bold dy over bold dx bold equals bold e to the power of bold x bold plus bold y end exponent bold minus bold e to the power of bold x bold therefore bold space bold left parenthesis bold e to the power of bold y bold minus bold e to the power of bold x bold. bold e to the power of bold x bold right parenthesis bold dy over bold dx bold equals bold e to the power of bold x bold. bold e to the power of bold y bold minus bold e to the power of bold x bold therefore bold space bold space bold space bold e to the power of bold y bold left parenthesis bold 1 bold minus bold e to the power of bold x bold right parenthesis bold dy over bold dx bold equals bold e to the power of bold x bold left parenthesis bold e to the power of bold y bold minus bold 1 bold right parenthesis bold therefore bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold dy over bold dx bold equals fraction numerator bold e to the power of bold x bold left parenthesis bold e to the power of bold y bold minus bold 1 bold right parenthesis over denominator bold e to the power of bold y bold left parenthesis bold 1 bold minus bold e to the power of bold x bold right parenthesis end fraction bold therefore bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold dy over bold dx bold equals bold minus fraction numerator bold e to the power of bold x bold left parenthesis bold e to the power of bold y bold minus bold 1 bold right parenthesis over denominator bold e to the power of bold y bold left parenthesis bold e to the power of bold x bold minus bold 1 bold right parenthesis end fraction

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