Limits and Derivatives

Limits and Derivatives

Question

Differentiate the following functions w.r.t. x


space space space space space space space space space space space space fraction numerator left parenthesis straight x plus 2 right parenthesis left parenthesis 1 minus 3 straight x right parenthesis over denominator 2 straight x plus 1 end fraction semicolon space space straight x space not equal to space fraction numerator negative 1 over denominator 2 end fraction

Answer

Let                 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space space equals space space space fraction numerator left parenthesis straight x plus 2 right parenthesis left parenthesis 1 minus 3 straight x right parenthesis over denominator 2 straight x plus 1 end fraction semicolon space straight x space not equal to fraction numerator negative 1 over denominator 2 end fraction space space space or space straight f left parenthesis straight x right parenthesis space space equals space fraction numerator negative 3 straight x squared minus 5 straight x plus 2 over denominator 2 straight x plus 1 end fraction

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                                                                                                                    (By Quotient Rule)

       space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis 2 straight x plus 1 right parenthesis left parenthesis negative 6 straight x minus 5 right parenthesis minus left parenthesis negative 3 straight x squared minus 5 straight x plus 2 right parenthesis left parenthesis 2 plus 0 right parenthesis over denominator left parenthesis 2 straight x plus 1 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 equals space space space fraction numerator negative 12 straight x squared minus 10 straight x minus 6 straight x minus 5 plus 6 straight x squared plus 10 straight x minus 4 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction equals fraction numerator negative 6 straight x squared minus 6 straight x minus 9 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction

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