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Continuity And Differentiability

Question
CBSEENMA12035399
Wired Faculty App

If space straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis equals 1 comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction equals open parentheses dy over dx close parentheses squared

Solution
Here space space space straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis equals 1 therefore space space space log left square bracket straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis right square bracket equals log space 1 therefore space space space log space straight e to the power of straight y left parenthesis straight x plus 1 right parenthesis equals 0 therefore space space space space space space space space space space straight y space log space straight e equals negative log left parenthesis straight x plus 1 right parenthesis therefore space space space space space space space space space space space space space space space space space space space space straight y equals negative log left parenthesis straight x plus 1 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 space space space space space space space space space 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 loge equals 1 right square bracket therefore space space space space space space space space space space space space space space dy over dx equals negative fraction numerator 1 over denominator straight x plus 1 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 space space space... left parenthesis 1 right parenthesis therefore space space space space space space space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative open square brackets fraction numerator left parenthesis straight x plus 1 right parenthesis. begin display style straight d over dx end style left parenthesis 1 right parenthesis minus 1. begin display style straight d over dx end style left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction close square brackets equals negative open square brackets fraction numerator left parenthesis straight x plus 1 right parenthesis.0 minus 1.1 over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction equals open parentheses negative fraction numerator 1 over denominator straight x plus 1 end fraction close parentheses squared therefore space space space space space space space space space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals open parentheses dy over dx 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 space space 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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