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Continuity And Differentiability

Question
CBSEENMA12034752
Wired Faculty App

If space straight y equals fraction numerator straight e to the power of straight x minus straight e to the power of negative straight x end exponent over denominator straight e to the power of straight x plus straight e to the power of negative straight x end exponent end fraction comma space prove space that space dy over dx equals 1 minus straight y squared.

Solution
straight y equals fraction numerator straight e to the power of straight x minus straight e to the power of negative straight x end exponent over denominator straight e to the power of straight x plus straight e to the power of negative straight x end exponent 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 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 fraction numerator left parenthesis bold e to the power of bold x bold plus bold e to the power of bold minus bold x end exponent right parenthesis begin display style straight d over dx end style left parenthesis straight e to the power of straight x minus straight e to the power of negative straight x end exponent right parenthesis minus left parenthesis straight e to the power of straight x minus straight e to the power of negative straight x end exponent right parenthesis begin display style straight d over dx end style left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis over denominator left parenthesis straight e to the power of straight x minus straight e to the power of negative straight x end exponent right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator left parenthesis bold e to the power of bold x bold plus bold e to the power of bold minus bold x end exponent right parenthesis left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis plus left parenthesis straight e to the power of straight x minus straight e to the power of negative straight x end exponent right parenthesis left parenthesis straight e to the power of straight x minus straight e to the power of negative straight x end exponent right parenthesis over denominator left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis squared minus left parenthesis straight e to the power of straight x minus straight e to the power of negative straight x end exponent right parenthesis squared over denominator left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis squared end fraction equals fraction numerator left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis squared over denominator left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis squared end fraction minus fraction numerator left parenthesis straight e to the power of straight x minus straight e to the power of negative straight x end exponent right parenthesis squared over denominator left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis squared end fraction therefore space dy over dx equals 1 minus straight y 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 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 space space space space space space space space space

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