Конспект лекций по математическому анализу. Шерстнев А.Н. - 56 стр.

  fORMULY DLQ DIFFERENCIALOW QWLQ@TSQ O^EWIDNYM SLEDSTWIEM SOOT-
WETSTWU@]IH FORMUL DLQ PROIZWODNYH. fORMULY DLQ PROIZWODNYH SLE-
DU@T IZ WY^ISLENIJ:
 (f + g)(x + h) , (f + g)(x) = [f (x + h) , f (x)] + [g(x + h) , g(x)]
                     =  f 0(x)h + o(h) + g0(x)h + o(h)
                     =  [f 0(x) + g0(x)]h + o(h) (h ! 0);
         (f 
 g)0(x) =  lim 1 [f (x + h)g(x + h) , f (x)g(x)]
                        h!0 h
                              f ( x  +   h ) , f  ( x )                    g ( x + h ) , g ( x ) 
                     =  lim
                        h!0               h             g(x) + f (x + h) 
          h
                     =  f 0(x)g(x) + f (x)g0(x); 
          (1=g)0(x) =   lim 1           1 , 1
                        h!0 h g (x + h) g (x)                                      g0(x) : >
                    = hlim    ,  1 [ g ( x +  h ) ,   g ( x)] 
      1          = ,
                         !0 h                                   g(x)g(x + h)        g2(x)
    2. [sLEDSTWIE]. (cf )0 (x) = cf 0 (x); c = const.
    3. [dIFFERENCIROWANIE SUPERPOZICII FUNKCIJ]. pUSTX f : E ! R;
g : F ! R; f (E )  F; f DIFFERENCIRUEMA W x 2 E , A g DIFFERENCIRUEMA
W f (x). tOGDA g  f DIFFERENCIRUEMA W x, PRI^EM
           (g  f )0(x) = g0(f (x))f 0(x); d(g  f )(x) = g0(f (x))df (x):
 dEJSTWITELXNO,
 g(f (x + h)) , g(f (x)) = g(f (x) + [f (x + h) , f (x)]) , g(f (x))
                             = g0(f (x))[f (x + h) , f (x)] + o(f (x + h) , f (x)):
tAK KAK o(f (x + h) , f (x)) = o(h) (h ! 0), IMEEM OTS@DA
          g(f (x + h)) , g(f (x)) = g0(f (x))f 0(x)h + o(h) (h ! 0): >
    4.[dIFFERENCIROWANIE OBRATNOJ FUNKCII]. pUSTX f I g | WZAIMNO
OBRATNYE FUNKCII. pUSTX g NEPRERYWNA W TO^KE x, A f DIFFERENCIRUE-
MA W TO^KE g(x), PRI^EM f 0(g(x)) 6= 0. tOGDA g DIFFERENCIRUEMA W TO^KE
xI
                             g0(x) = f 0(g1(x)) :

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