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 space space space space space space space space space space space space space fraction numerator straight x minus 1 over denominator 3 straight x cubed plus 2 end fraction comma space straight x space not equal to space cube root of fraction numerator negative 2 over denominator 3 end fraction end root

Answer

Let     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 equals space fraction numerator straight x minus 1 over denominator 3 straight x cubed plus 2 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 straight f left parenthesis straight x right parenthesis space space equals space fraction numerator left parenthesis 3 straight x cubed plus 2 right parenthesis begin display style straight d over dx end style left parenthesis straight x minus 1 right parenthesis minus left parenthesis straight x minus 1 right parenthesis begin display style straight d over dx end style left parenthesis 3 straight x cubed plus 2 right parenthesis over denominator open parentheses 3 straight x cubed plus 2 close parentheses squared end fraction        (By Quotient Rule)
          space space space space space space space space space space space space space space space space space space space equals space space space space fraction numerator left parenthesis 3 straight x cubed plus 2 right parenthesis left parenthesis 1 minus 0 right parenthesis minus left parenthesis straight x minus 1 right parenthesis left parenthesis 9 straight x squared plus 0 right parenthesis over denominator left parenthesis 3 straight x cubed plus 2 right parenthesis squared end fraction equals fraction numerator 3 straight x cubed plus 2 minus 9 straight x cubed plus 9 straight x squared over denominator left parenthesis 3 straight x cubed plus 2 right parenthesis squared end fraction


                space space space space space space space space space space space space space equals space space fraction numerator negative 6 straight x cubed plus 9 straight x squared plus 2 over denominator left parenthesis 3 straight x cubed plus 2 right parenthesis squared end fraction

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