Microsoft Word - Derivative Cheat Sheet PDF

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DerivativeCheatSheet Definition If  󰇛󰇜thenthe(first‐order)derivativeisdefinedtobe′󰇛󰇜  lim  



denotedas 󰆒 󰇛󰇜,  󰆒 ,  ,  ,  󰇛󰇜󰇛󰇜.

→

󰇛󰇜󰇛󰇜 ,whichisoften 

Interpretation   ′󰇛󰇜istheslopeofthetangentlineto  󰇛󰇜at  andtheequationforthetangent lineatat  isat  󰇛󰇜  ′󰇛󰇜󰇛  󰇜.Alternatively,thederivativeisalsotheinstantaneous rateofchangeof󰇛󰇜at   Basicrules Forthebasicderivativerulesbelow,f(x)andg(x)aredifferentiablefunctions(ie.Thederivative exists),a,candnareanyrealnumbers,eisthebaseofthenaturallogarithmln.  󰆒

󰇛󰇜   󰆒 󰇛 󰇜 󰆒

󰇛󰇜  󰇛󰇜  󰆒 󰇛󰇜   󰆒 󰇛󰇜 󰇛󰇜󰆒   󰆒   ′





‐Productrule

 󰆒  󰆒   󰆒      

 󰇛󰇜  0 

 󰇛  󰇜  󰇛󰇜  

1  1    󰇛 󰇜    

 󰇛󰇛󰇛󰇜󰇜 



 



  󰆒 󰇛󰇜󰆒 󰇛󰇜 

‐Chainrule

󰇛  󰇜   ln󰇛󰇜 󰇛  󰇜    

1  󰇛ln󰇛󰇜󰇜  ,   0  

1  󰇛log 󰇜  ,   0    Higherorderderivatives Thesecondorderderivativeisdefinedasthederivativeofthefirstderivative 󰆒󰆒 󰇛󰇜  󰇛′ 󰇛 󰇜󰇜′,and denotedas

 󰆒󰆒 󰇛󰇜   󰇛󰇜 󰇛󰇜 

   

Thenthorderderivativeisdefinedasthederivativeofthe(n‐1)stderivative  󰆒

 󰇛󰇜 󰇛󰇜  󰇡 󰇛󰇜 󰇛 󰇜󰇢 anddenotedas 󰇛󰇜 󰇛󰇜         

 

 

...