Mathc initiation/a0074

Sommaire

Calculons la dérivée :

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


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


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

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

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


 sin(x)' =  cos(x)

Remarque :

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