Mathc initiation/a0073

Sommaire

Calculons la dérivée :

                  cos(x+h) - cos(x)
  cos(x)' = lim   -----------------  
            h->0           h
       
       
                 (cos(x)cos(h)- sin(x)sin(h)) - cos(x)
 cos(x)' = lim    ------------------------------------  
           h->0                  h       


                 (cos(x)cos(h) - cos(x)) - sin(x)sin(h)
 cos(x)' = lim    --------------------------------------  
           h->0                  h       


                  cos(x) (cos(h)-1)            sin(x)sin(h)
 cos(x)' = lim    ----------------- -    lim   ------------  
           h->0         h                h->0     h
       
       
                          (cos(h)-1)                   sin(h)
 cos(x)' =   cos(x) lim   ----------  -  sin(x)  lim   ------ 
                    h->0      h                  h->0    h

                        
 cos(x)' =   cos(x) 0 - sin(x) 1  
                                  


 cos(x)' =  -sin(x)

Remarque :

• lim h->0 sin(h)/h = 1
• lim h->0 (1-cos(h))/h = 0