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Continuity And Differentiability

Question
CBSEENMA12034774
Wired Faculty App

If space straight x equals fraction numerator 1 plus log space straight t over denominator straight t squared end fraction comma space straight y equals fraction numerator 3 plus 2 space log space straight t over denominator straight t end fraction comma space straight t greater than 0 comma space prove space that space straight y dy over dx minus 2 straight x open parentheses dy over dx close parentheses squared equals 1

Solution
Here space straight x equals fraction numerator 1 plus log space straight t over denominator straight t squared end fraction comma space straight y equals fraction numerator 3 plus 2 space log space straight t over denominator straight t end fraction therefore space dx over dt equals fraction numerator left parenthesis straight t squared right parenthesis. begin display style straight d over dt end style left parenthesis 1 plus log space straight t right parenthesis minus left parenthesis 1 plus log space straight t right parenthesis. begin display style straight d over dt end style left parenthesis straight t squared right parenthesis over denominator left parenthesis straight t squared right parenthesis squared end fraction equals fraction numerator straight t squared. begin display style 1 over straight t end style minus left parenthesis 1 plus log space straight t right parenthesis.2 straight t over denominator straight t to the power of 4 end fraction equals fraction numerator 1 minus 2 minus 2 space lg space straight t over denominator straight t cubed end fraction equals fraction numerator negative 1 minus 2 space log space straight t over denominator straight t cubed end fraction therefore space dx over dt equals negative fraction numerator 1 plus 2 space log space straight t over denominator straight t cubed 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... left parenthesis 1 right parenthesis Also space dy over dt equals fraction numerator straight t. begin display style straight d over dt end style left parenthesis 3 plus 2 space log space straight t right parenthesis minus left parenthesis 3 plus 2 space log space straight t right parenthesis. begin display style straight d over dt end style left parenthesis straight t right parenthesis over denominator straight t squared end fraction space space space space space space space space space space space space space space space space space equals fraction numerator straight t. begin display style 2 over straight t end style minus left parenthesis 3 plus 2 space log space straight t right parenthesis.1 over denominator straight t squared end fraction equals fraction numerator 2 minus 3 minus 2 space log space straight t over denominator straight t squared end fraction equals fraction numerator negative 1 minus 2 space log space straight t over denominator straight t squared end fraction therefore space dy over dt equals negative fraction numerator 1 plus 2 space log space straight t over denominator straight t squared 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... left parenthesis 2 right parenthesis Now space dy over dx equals fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction equals open parentheses negative fraction numerator 1 plus 2 space log space straight t over denominator straight t squared end fraction space close parentheses cross times open parentheses negative fraction numerator straight t cubed over denominator 1 plus 2 space log space straight t end fraction close parentheses 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 left parenthesis 2 right parenthesis right square bracket therefore space dy over dx equals straight t space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 equals straight y dy over dx minus 2 straight x open parentheses dy over dx close parentheses squared space space space space space space space space space space space equals open parentheses fraction numerator 3 plus 2 space log over denominator straight t end fraction close parentheses left parenthesis straight t right parenthesis minus 2 open parentheses fraction numerator 1 plus space log space straight t over denominator straight t squared end fraction close parentheses left parenthesis straight t squared 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 left square bracket because space of space left parenthesis 3 right parenthesis right square bracket space space space space space space space space space space space equals left parenthesis 3 plus 2 space log space straight t right parenthesis minus 2 left parenthesis 1 plus log space straight t right parenthesis equals 3 plus 2 log space straight t minus 2 minus 2 space log space space space space space space space space space space space equals 1 equals straight R. straight H. straight S.

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