Конспект лекций по математическому анализу. Шерстнев А.Н. - 57 стр.

 tAK KAK g NEPRERYWNA W x, WELI^INA g(x + h) , g(x) MALA, ESLI MALO
SME]ENIE h. pO\TOMU SPRAWEDLIWA WYKLADKA
          h = x + h , x = f (g(x + h)) , f (g(x))
             = f (g(x) + g(x + h) , g(x)) , f (g(x))
             = f 0(g(x))(g(x + h) , g(x)) + o(g(x + h) , g(x))
             = (g(x + h) , g(x))(f 0(g(x)) + o(1)) (h ! 0);
sLEDOWATELXNO,
          lim 1 [g(x + h) , g(x)] = lim [f 0(g(x)) + o(1)],1 = 1 : >
          h!0 h                     h!0                       f 0(g(x))
     tABLICA PROIZWODNYH \LEMENTARNYH FUNKCIJ].~ASTX PRIWEDENNYH
   5. [
NIVE FORMUL POLU^ENA RANEE. oSTALXNYE POLU^A@TSQ S POMO]X@ DOKA-
ZANNYH WY[E UTWERVDENIJ (PP. 1 { 4). dADIM NESKOLXKO ILL@STRACIJ.
   pOLOVIM f (x) = ax (x 2 R); g(x) = loga x (x > 0). sOGLASNO P. 4
                (loga x)0 = (aloga x 
 ln a),1 = 1 (x > 0):
                                                 x ln a
w SILU P. 3 I 29.13 (ln jxj)0 = jx1j 
 sgn x = 1=x (x 6= 0).
   pOLOVIM f (x) = sin x (jxj < 2 ); g(x) = arcsin x (jxj < 1). tOGDA (P. 4)
                (arcsin x)0 =        1          = p 1 2 (jxj < 1):
                               cos(arcsin x)       1,x
   fORMULA (xb)0 = bxb,1 (x 2 R) LEGKO POLU^AETSQ PO INDUKCII DLQ
b = 0; 1; 2; : : : . eSLI b = ,1; ,2; : : :, TO
                (xb)0 = (1=x,b )0 = ,(x,b)0=x,2b = bxb,1 (x 6= 0):
eSLI, NAKONEC, b PROIZWOLXNO, TO FORMULA (xb)0 = bxb,1 (x > 0) ESTX
SLEDSTWIE PREDSTAWLENIQ xb = eb ln x.
    6. z A M E ^ A N I E. dLQ WY^ISLENIQ PROIZWODNYH FUNKCIJ WI-
DA f (x) = u(x)v(x) (u(x) > 0) SLEDUET WOSPOLXZOWATXSQ PREDSTAWLENIEM
f (x) = ev(x)ln u(x).
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