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Fórmules de derivades

Derivades d'arrels

Tingueu en compte que les fórmules 

    bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals square root of bold x bold space bold space bold spacebold space bold rightwards arrow bold space bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold 1 over denominator bold 2 square root of bold x end fraction

   bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals square root of bold u bold left parenthesis bold x bold right parenthesis end root bold space bold space bold space bold space bold space bold space bold rightwards arrow bold space bold space bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold 1 over denominator bold 2 square root of bold u bold left parenthesis bold x bold right parenthesis end root end fraction bold times bold italic u bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold u bold apostrophe bold left parenthesis bold x bold right parenthesis over denominator bold 2 square root of bold u bold left parenthesis bold x bold right parenthesis end root end fraction

   només són vàlides per a les arrels quadrades

   Com fem la derivada d'una arrel no quadrada?  Tenint en compte:

    n-th root of x equals x to the power of 1 divided by n end exponent n-th root of x to the power of m end root equals x to the power of m divided by n end exponent

    Exemples

  •  y equals fifth root of x space space rightwards arrow y equals x to the power of 1 divided by 5 end exponent space space rightwards arrow y apostrophe equals 1 fifth x to the power of 1 fifth minus 1 end exponent equals 1 fifth x to the power of negative 4 divided by 5 end exponent equals fraction numerator 1 over denominator 5 times x to the power of 4 divided by 5 end exponent end fraction equals fraction numerator 1 over denominator 5 fifth root of x to the power of 4 end root end fraction
  •   y equals cube root of x squared end root space space rightwards arrow y equals x to the power of 2 divided by 3 end exponent space space rightwards arrow y apostrophe equals 2 over 3 x to the power of 2 over 3 minus 1 end exponent equals 2 over 3 x to the power of negative 1 divided by 3 end exponent equals fraction numerator 2 over denominator 3 times x to the power of 1 divided by 3 end exponent end fraction equals fraction numerator 2 over denominator 3 cube root of x end fraction
  •   y equals cube root of x squared plus 3 x end root space space rightwards arrow y equals open parentheses x squared plus 3 x close parentheses to the power of 1 divided by 3 end exponent space space

                  y apostrophe equals 1 third open parentheses x squared plus 3 x close parentheses to the power of 1 third minus 1 end exponent times left parenthesis 2 x plus 3 right parenthesis equals 1 third open parentheses x squared plus 3 x close parentheses to the power of negative 2 over 3 end exponent times left parenthesis 2 x plus 3 right parenthesis equals fraction numerator 2 x plus 3 over denominator 3 cube root of open parentheses x squared plus 3 x close parentheses squared end root end fraction

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