Конспект лекций по математическому анализу. Шерстнев А.Н. - 58 стр.

(sin x)0 = cos x (x 2 R)            (ax)0 = ax ln a (x 2 R)
(cos x)0 = , sin x (x 2 R)          (ex)0 = ex (x 2 R)
(tg x)0 = cos12 x (cos x 6= 0)      (ln jxj)0 = 1=x (x 6= 0)
(ctg x)0 = , sin12 x (sin x 6= 0)   (loga jxj)0 = x ln1 a
(arcsin x)0 = q 1 2 (jxj < 1)                 (x 6= 0; 0 < a 6= 1)
                 1,x
(arccos x) = , p 1 2 (jxj < 1)
            0                       (sh x)0 = ch x (x 2 R)
                    1,x
          0      1
(arctg x) = 1 + x2 (x 2 R)          (ch x)0 = sh x (x 2 R)
(arcctg x)0 = , 1 +1 x2 (x 2 R)     (th x)0 = 12 (x 2 R)
                                              ch x1
(xb)0 = bxb,1 (x > 0)               (cth x) = , 2 (x 6= 0)
                                            0
                                                 sh x

     u P R A V NpE N I Q. nAJTI PROIZWODNYE FUNKCIJ:
     7. ln(x + a2 + x2),
     8. arcsin x a       (OTWET: (a2 , x2),1=2 
 sgn a),
     9. arcsin x 1       (OTWET: ,[x(x2 , 1),1=2] 
 sgn x (jxj > 1)),
     10. xx ,
     11. ln j tg xj,
                     expf,1=x2g; ESLI x 6= 0,
     12. f (x) =
                      0;               ESLI x = 0.
     x31. pROIZWODNYE WYS[IH PORQDKOW
     1. pUSTX f : E ! R DIFFERENCIRUEMA W KAVDOJ TO^KE E , TO ESTX
OPREDELENA FUNKCIQ f 0 : E ! R . eSLI f 0 DIFFERENCIRUEMA W TO^-
KE x0, TO ^ISLO (f 0)0(x0) NAZYWAETSQ 2-J PROIZWODNOJ f W TO^KE x0 I
                                2 f (x0)
OBOZNA^AETSQ f 00(x0) ILI d dx       2 . pUSTX, W ^ASTNOSTI, f DIFFERENCI-
                                                                  0
RUEMA W KAVDOJ TO^KE MNOVESTWA E . tOGDA NA E OPREDELENA FUNKCIQ
f 00(x)  (f 0)0(x) (x 2 E ), KOTORAQ NAZYWAETSQ 2-J PROIZWODNOJ FUNKCII f

                                     58