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Continuity And Differentiability

Question
CBSEENMA12034768
Wired Faculty App

Find space dy over dx space where space straight x equals fraction numerator straight a left parenthesis 1 minus straight t squared right parenthesis over denominator 1 minus straight t squared end fraction comma space straight y space equals fraction numerator 2 bt over denominator 1 minus straight t squared end fraction

Solution
Here space space straight x equals fraction numerator straight a left parenthesis 1 minus straight t squared right parenthesis over denominator 1 minus straight t squared end fraction equals straight a open square brackets fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction close square brackets dx over dt equals straight a open square brackets fraction numerator left parenthesis 1 plus straight t squared right parenthesis begin display style straight d over dt end style left parenthesis 1 minus straight t squared right parenthesis minus left parenthesis 1 minus straight t squared right parenthesis. begin display style straight d over dt end style left parenthesis 1 plus straight t squared right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets space space space space space space space space equals straight a open square brackets fraction numerator left parenthesis 1 plus straight t squared right parenthesis left parenthesis negative 2 straight t right parenthesis minus left parenthesis 1 minus straight t squared right parenthesis left parenthesis 2 straight t right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets equals straight a open square brackets fraction numerator negative 2 straight t minus 2 straight t cubed minus 2 straight t plus 2 straight t cubed over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets space space space space space space space space equals straight a open square brackets fraction numerator negative 4 straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets therefore dx over dt equals negative fraction numerator 4 at over denominator left parenthesis 1 plus straight t squared right parenthesis 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... left parenthesis 1 right parenthesis Also space straight y space equals fraction numerator 2 bt over denominator 1 minus straight t squared end fraction equals 2 straight b open square brackets fraction numerator straight t over denominator 1 plus straight t squared end fraction close square brackets dy over dt equals 2 straight b open square brackets fraction numerator left parenthesis 1 plus straight t squared right parenthesis begin display style straight d over dt end style left parenthesis straight t right parenthesis minus straight t begin display style straight d over dt end style left parenthesis 1 plus straight t squared right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets equals 2 straight b open square brackets fraction numerator left parenthesis 1 plus straight t squared right parenthesis.1 minus straight t space.2 straight t over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction close square brackets therefore space dy over dt equals fraction numerator 2 straight b left parenthesis 1 minus straight t squared right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis 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... 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 2 straight b left parenthesis 1 minus straight t squared right parenthesis over denominator left parenthesis 1 plus straight t squared right parenthesis squared end fraction cross times fraction numerator left parenthesis 1 plus straight t squared right parenthesis squared over denominator negative 4 at end fraction equals negative fraction numerator straight b left parenthesis 1 minus straight t squared right parenthesis over denominator 2 at end fraction.

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