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Continuity And Differentiability

Question
CBSEENMA12034753
Wired Faculty App

If space straight y equals fraction numerator 3 minus straight u over denominator 2 plus straight u end fraction comma straight u equals fraction numerator 4 straight x over denominator 1 minus straight x squared end fraction space find space dy over dx

Solution
Here space straight y equals fraction numerator 3 minus straight u over denominator 2 plus straight u end fraction therefore space dy over dx equals fraction numerator left parenthesis 2 plus straight u right parenthesis begin display style straight d over du end style left parenthesis 3 minus straight u right parenthesis minus left parenthesis 3 minus straight u right parenthesis begin display style straight d over du end style left parenthesis 2 plus straight u right parenthesis over denominator left parenthesis 2 plus straight u right parenthesis squared end fraction equals fraction numerator left parenthesis 2 plus straight u right parenthesis left parenthesis negative 1 right parenthesis minus left parenthesis 3 minus straight u right parenthesis.1 over denominator left parenthesis 2 plus straight u right parenthesis squared end fraction space space space space space space space space space space space space equals fraction numerator 2 minus straight u minus 3 plus straight u over denominator left parenthesis 2 plus straight u right parenthesis squared end fraction therefore space dy over dx equals negative fraction numerator 5 over denominator left parenthesis 2 plus straight u 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 space space space space space space space space space space... left parenthesis 1 right parenthesis Also space straight u equals fraction numerator 4 straight x over denominator 1 minus straight x squared end fraction equals 4 open square brackets fraction numerator straight x over denominator 1 minus straight x squared end fraction close square brackets therefore space du over dx equals 4 open curly brackets fraction numerator left parenthesis 1 minus straight x squared right parenthesis begin display style straight d over dx end style left parenthesis straight x right parenthesis minus straight x. begin display style straight d over dx end style left parenthesis 1 minus straight x squared right parenthesis over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction close curly brackets space space space space space space space space space space space space space equals 4 open square brackets fraction numerator left parenthesis 1 minus straight x squared right parenthesis.1 minus straight x. left parenthesis negative 2 straight x right parenthesis over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction close square brackets equals 4 open square brackets fraction numerator 1 minus straight x squared plus 2 straight x squared over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction close square brackets therefore space du over dx equals fraction numerator 4 left parenthesis 1 minus straight x squared right parenthesis over denominator left parenthesis 1 minus straight x 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 space space space space space space space space space space space space space... left parenthesis 2 right parenthesis Now space dy over dx equals dy over du. du over dx equals negative fraction numerator 5 over denominator left parenthesis 2 plus straight u right parenthesis squared end fraction. fraction numerator 4 left parenthesis 1 plus straight x squared right parenthesis over denominator left parenthesis 1 minus straight x 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 left square bracket because space of space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis right square bracket space space space space space space space space space space space space space space space space space equals fraction numerator negative 5 left parenthesis 1 minus straight x squared right parenthesis squared over denominator 4 left parenthesis 1 plus 2 straight x minus straight x squared right parenthesis squared end fraction. fraction numerator 4 left parenthesis 1 plus straight x squared right parenthesis over denominator left parenthesis 1 minus straight x squared right parenthesis squared end fraction equals negative fraction numerator 5 left parenthesis 1 minus straight x squared right parenthesis squared over denominator left parenthesis 1 plus 2 straight x minus straight x squared right parenthesis squared end fraction

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