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Continuity And Differentiability

Question

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Differentiate $square root of 3 straight x squared plus 2 straight x end root$ w.r.t x.

Solution

Let space space space space space space straight y equals space fraction numerator 3 straight x squared plus 2 straight x over denominator square root of 3 straight x squared plus 2 straight x end root end fraction therefore space dy over dx equals fraction numerator square root of 3 straight x squared plus 2 straight x end root begin display style straight d over dx end style left parenthesis 3 straight x plus 1 right parenthesis minus left parenthesis 3 straight x plus 1 right parenthesis begin display style straight d over dx end style open parentheses square root of 3 straight x squared plus 2 straight x end root close parentheses over denominator 3 straight x squared plus 2 straight x end fraction equals fraction numerator square root of 3 straight x squared plus 2 straight x end root space 3 minus left parenthesis 3 straight x plus 1 right parenthesis begin display style fraction numerator 6 straight x plus 2 over denominator 2 square root of 3 straight x squared plus 2 straight x end root end fraction end style over denominator 3 straight x squared plus 2 straight x end fraction equals fraction numerator 9 straight x squared plus 6 straight x minus 9 straight x squared minus 6 straight x minus 1 over denominator left parenthesis 3 straight x squared plus 2 straight x right parenthesis to the power of bevelled 3 over 2 end exponent end fraction equals negative fraction numerator 1 over denominator left parenthesis 3 straight x squared plus 2 straight x right parenthesis to the power of bevelled 3 over 2 end exponent end fraction

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