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Fórmules de derivades

Derivada de funcions compostes

Per derivar una funció composta s'aplica la Regla de la cadena: 

     f left parenthesis x right parenthesis equals u left parenthesis v left parenthesis x right parenthesis right parenthesis space space space rightwards arrow space space space f apostrophe left parenthesis x right parenthesis equals u apostrophe left parenthesis v left parenthesis x right parenthesis right parenthesis times v apostrophe left parenthesis x right parenthesis       

però a la pràctica el que fem és  aplicar les fórmules de derivades. 

Fórmules:

 bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic k bold space bold space bold space bold space bold italic k bold element of bold real numbers bold space bold space bold space

bold space bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold 0

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic k bold times bold italic u bold left parenthesis bold italic x bold right parenthesis bold space bold space bold space bold italic k bold element of bold real numbers bold space bold space bold space

bold space bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold italic k bold times bold italic u bold apostrophe bold left parenthesis bold italic x bold right parenthesis

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis to the power of bold n bold space bold space bold space bold space bold space                   

bold space bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold italic n bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis to the power of bold n bold minus bold 1 end exponent bold times bold italic u bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold space bold space bold space bold space bold space bold space bold space 

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 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

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic e to the power of bold u bold left parenthesis bold x bold right parenthesis end exponent bold space bold space bold space

bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold italic e to the power of bold u bold left parenthesis bold x bold right parenthesis end exponent bold times bold italic u bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold space bold space bold space

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic a to the power of bold u bold left parenthesis bold x bold right parenthesis end exponent bold space bold space bold space

 bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold italic a to the power of bold u bold left parenthesis bold x bold right parenthesis end exponent bold times bold ln bold space bold italic a bold space bold times bold italic u bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold space bold space

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic l bold italic n bold space bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis

 bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold 1 over denominator bold u bold left parenthesis bold x bold right parenthesis 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 u bold left parenthesis bold x bold right parenthesis end fraction

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold log subscript bold a bold space bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis

 bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold 1 over denominator bold ln bold space bold a end fraction bold times fraction numerator bold 1 over denominator bold u bold left parenthesis bold x bold right parenthesis end fraction bold times bold italic u bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold 1 over denominator bold ln bold space bold a end fraction bold times fraction numerator bold u bold apostrophe bold left parenthesis bold x bold right parenthesis over denominator bold u bold left parenthesis bold x bold right parenthesis end fraction
 bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold sin bold space bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold cos bold space bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis bold times bold italic u bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold space
 bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold cos bold space bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis  bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold minus bold italic s bold italic i bold italic n bold space bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis bold times bold italic u bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold space
bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold italic t bold italic g bold left parenthesis bold italic u bold left parenthesis bold italic x bold right parenthesis bold right parenthesis  bold italic f 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 c bold o bold s to the power of bold 2 bold left parenthesis bold u bold left parenthesis bold x bold right parenthesis bold right parenthesis end fraction

Aquí podeu veure uns quants exemples:  

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