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Fórmules de derivades

Funció derivada i operacions

Derivada de la suma.

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic g bold left parenthesis bold italic x bold right parenthesis bold plus bold italic h bold left parenthesis bold italic x bold right parenthesis bold space bold space bold rightwards arrow bold space bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold italic g bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold plus bold italic h bold apostrophe bold left parenthesis bold italic x bold right parenthesis space

Exemple.

f left parenthesis x right parenthesis equals e to the power of 2 x end exponent plus x cubed space space rightwards arrow space space f left parenthesis x right parenthesis equals 2 times e to the power of 2 x end exponent plus 3 x squared

                  

Derivada de la diferència.

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic g bold left parenthesis bold italic x bold right parenthesis bold minus bold italic h bold left parenthesis bold italic x bold right parenthesis bold space bold space bold rightwards arrow bold space bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold italic g bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold minus bold italic h bold apostrophe bold left parenthesis bold italic x bold right parenthesis space

Exemple.

f left parenthesis x right parenthesis equals 5 x minus ln space x space space rightwards arrow space space f apostrophe left parenthesis x right parenthesis equals 5 minus 1 over x

Derivada del producte

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic g bold left parenthesis bold italic x bold right parenthesis bold times bold italic h bold left parenthesis bold italic x bold right parenthesis bold space bold space bold rightwards arrow bold space bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold italic g bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold times bold italic h bold left parenthesis bold italic x bold right parenthesis bold plus bold italic g bold left parenthesis bold italic x bold right parenthesis bold times bold italic h bold apostrophe bold left parenthesis bold italic x bold right parenthesis space

Exemple.

f left parenthesis x right parenthesis equals x times e to the power of 3 x end exponent space space rightwards arrow space space f left parenthesis x right parenthesis equals 1 times e to the power of 3 x end exponent plus x times e to the power of 3 x end exponent times 3 equals e to the power of 3 x end exponent left parenthesis 1 plus 3 x right parenthesis

Derivada del quocient

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold g bold left parenthesis bold x bold right parenthesis over denominator bold h bold left parenthesis bold x bold right parenthesis end fraction bold space bold 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 g bold apostrophe bold left parenthesis bold x bold right parenthesis bold times bold h bold left parenthesis bold x bold right parenthesis bold minus bold g bold left parenthesis bold x bold right parenthesis bold times bold h bold apostrophe bold left parenthesis bold x bold right parenthesis over denominator open parentheses bold h bold left parenthesis bold x bold right parenthesis close parentheses to the power of bold 2 end fraction

Exemple.

f left parenthesis x right parenthesis equals fraction numerator 5 x plus 2 over denominator 3 x minus 1 end fraction space space rightwards arrow space space f apostrophe left parenthesis x right parenthesis equals fraction numerator 5 times left parenthesis 3 straight x minus 1 right parenthesis minus left parenthesis 5 straight x plus 2 right parenthesis times 3 over denominator left parenthesis 3 x minus 1 right parenthesis squared end fraction equals fraction numerator 15 x minus 5 minus 15 x minus 6 over denominator left parenthesis 3 x minus 1 right parenthesis squared end fraction equals fraction numerator negative 11 over denominator left parenthesis 3 x minus 1 right parenthesis squared end fraction

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