Limits and Derivatives

Limits and Derivatives

Question

Find the derivative of space fraction numerator 1 over denominator square root of 3 straight x plus 1 end root end fraction at x =2.

Answer
space space space space space space space space space space space space space space space space space 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 2 right parenthesis space equals space space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis 2 plus straight h right parenthesis minus straight f left parenthesis 2 right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator square root of 3 left parenthesis 2 plus straight h right parenthesis plus 1 end root end fraction minus fraction numerator 1 over denominator square root of 3 left parenthesis 2 right parenthesis end root plus 1 end fraction end style over denominator straight h end fraction
                              space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator square root of 7 plus 3 straight h end root end fraction end style minus begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator square root of 7 minus square root of 7 plus 3 straight h end root over denominator square root of 7 space space square root of 7 plus 3 straight h end root end fraction close square brackets

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#6 {main}</pre>
                  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 limit as straight h rightwards arrow 0 of 1 fourth. fraction numerator 7 minus left parenthesis 7 plus 3 straight h right parenthesis over denominator square root of 7 square root of 7 plus 3 straight h end root left parenthesis square root of 7 plus square root of 7 plus 34 right parenthesis end root end fraction

                           space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator negative 3 over denominator square root of 7 plus square root of 7 plus 3 straight h end root left parenthesis square root of 7 plus square root of 7 plus 3 straight h right parenthesis end root end fraction equals fraction numerator negative 3 over denominator square root of 7 square root of 7 left parenthesis square root of 7 plus square root of 7 right parenthesis end fraction equals fraction numerator negative 3 over denominator 14 square root of 7 end fraction

                            


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