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Continuity And Differentiability

Question

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$Here space straight x equals straight e to the power of straight t. space log space straight t therefore space dx over dt equals straight e to the power of straight t. straight d over dt left parenthesis log space straight t right parenthesis plus log space straight t. straight d over dt left parenthesis straight e to the power of straight t right parenthesis equals straight e to the power of straight t.1 over straight t plus log space straight t. straight e to the power of straight t equals straight e to the power of straight t open parentheses 1 over straight t plus log space straight t close parentheses therefore dx over dt equals straight e to the power of straight t open parentheses fraction numerator 1 plus straight t space log space straight t over denominator straight t end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 straight y equals straight t. space log space straight t therefore space dy over dt equals straight t. straight d over dt left parenthesis log space straight t right parenthesis plus log space straight t space straight d over dt left parenthesis straight t right parenthesis equals straight t.1 over straight t plus log space straight t.1 therefore space dy over dt equals 1. space log space 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... 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 fraction numerator straight t left parenthesis 1. space log space straight t right parenthesis over denominator straight e to the power of straight t left parenthesis 1 plus straight t space log space straight t right parenthesis 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 left square bracket because space of space left parenthesis 1 right parenthesis comma left parenthesis 2 right parenthesis right square bracket$

Solution

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