,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x) x a, R a R, (), a R, x a f (a)., R a R, f(x)
, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f dx n,., IB ( 4 ), f(x), a R, a R, (),,,, x R,..,, f(x) x 0, f(x).,,, f (x) lim 0 f(x + ) f(x) lim 0 0 lim 0 lim 0 0 0, () 0,, dn f dx n d dx df dx d d n f dx n dx ` df dx d f dx, f(x),, f(x) dn f dx n
3, f(x) x.,, f (x) lim 0 f(x + ) f(x) lim 0 (x + ) x lim 0 lim 0, (x) 4, f(x) x.,, f f(x + ) f(x) (x) lim 0 (x + ) x lim 0 (x + x + ) x lim 0 x + lim 0 lim(x + ) 0 x, (x ) x, n N, f(x) x n, f(x + ) (x + ) n, (x + ) n x n + nx n +, n(n ) x n + + nx n + n (x + ) n x n nx n + n(n ) x n + + nx n + n 3, f(x), x R, 0. 4, f(x) x, x R,. 3
, f f(x + ) f(x) (x) lim 0 (x + ) n x n lim 0 lim 0 nx n { nx n + } n(n ) x n + + nx n + n, x n (x n ) nx n (), x n, x n,, x.3 f(x) sin x, cos x,, f(x + ) sin(x + ), cos(x + ), f(x) sin x, sin(x + ) sin x sin x cos + cos x sin sin x sin x (cos ) + cos x sin, 5 (sin x) sin(x + ) sin x lim 0 lim 0, f(x) cos x, { sin x cos + cos x sin cos sin sin x lim + cos x lim 0 0 } (3), 6 cos(x + ) cos x cos x cos sin x sin cos x cos x (cos ) sin x sin (cos x) lim 0 cos(x + ) cos x 5, sin x cos x 6, cos x sin x 4
{ lim cos x cos sin x sin } 0 cos sin cos x lim sin x lim 0 0, sin x, cos x, sin lim 0 lim 0 cos 7,,.,, O, A, B, C, D ( ). (4) (5) (6) y A C sin O cos tan B D x cos : sin, cos,,, AB, BD, AD., AB sin, AD,,, (5), sin lim 0 (7),,,, (7), OAB OCD,, OAD,.,, ( OAB ) ( OAD ) ( OCD ) 7, sin 0 0, cos 0,,, f(x) sin x x 0, f(x) cos x x 0. 5
, 8, cos sin tan (8) tan sin cos, (8),, sin cos sin cos, (9), (9), 0, sin lim 0 (0), (6),,,,,., BD BD cos, ABD,, cos BD (AD ) ( AB ) (), (), cos (AD ) ( ) AB () 9,, AB AD AD, lim 0 ( ) AB, lim 0 ( ) AD (3) 8, OAD,,, π,, π.,, ` π π, π.,, π,, π 9 π π, 0, < 0, (), cos s «AD «AB 6
., (), (3), cos lim 0 (4) 0 0, (6), (4), ( cos cos ), cos sin,,, cos cos ( ) cos sin sin ( cos ) sin sin sin cos sin sin sin sin sin (5), (5), { cos lim lim sin } sin 0 0 lim sin 0 lim 0 sin sin k lim lim sin k k 0 k k 0 (k. ) ( ) 0 ((0) ), (5), (6),, 0 (6) 0 (4), y cos x x 0 ( ), cos x, (4) 7
sin cos lim, lim 0 (7) 0 0, (3), (4) (7), (sin x) cos x, (cos x) sin x (8).4 0 < a R, f(x) a x,, a x+ a x a (9), (9),, f(x + ) f(x) a x+ a x a x a a x a x (a ) (a x ) a x+ a x lim 0 { } lim a x a 0 a x lim 0 a (0), f(0) a 0, a f(0 + ) f(0) lim lim 0 0 f (0), (0), (a x ) Ca x (), y a x x 0 C R, y sin x x 0, y cos x x 0,,, 0 8
., f(x) a x, C, a, y a x x 0 C,, 0 < a < C < 0 a C 0 < a C > 0 ( 3 )., a, C y y 3 x y x y ( )x 0 y x x 3:. a, x 0., (),, C, C a, e,, e lim 0, e. 3, a e, (), ( ) (e x ) e x (),, 3,, y x x 0, y 3 x x 0 3, e,, Taylor, f(x) e x, f() e, e.788 9
0 < a R, λ R, a x e λx (3), 4,, f(x) e x., f(x) x sin x f(x) e x +,,,,.,,.., g(x), (x), f(x) g(x) + (x),,, lim!0 g(x + ) g(x) (x + ) (x), lim!0 (4), f(x) (4) 5, (4), f(x), f(x + ) f(x) {g(x + ) + (x + )} {g(x) + (x)} {g(x + ) g(x)} + {(x + ) (x)} g(x + ) g(x) (x + ) (x) + (5), (5), 0, 4 3., λ log a, x R, (3). 5, (x),,, 0
(g(x) + (x)) g (x) + (x) (6),., (x + x 4 ) (x) + (x 4 ) ((6) ) + 4x 3 (() ), f(x), f(x) g (x) + g (x) + g (x),,,, (6), f(x) {g (x) + g (x)} + g 3 (x) [g (x) + g (x) + g 3 (x)] [{g (x) + g (x)} + g 3 (x)] {g (x) + g (x)} + g 3(x) ((6) ) {g (x) + g (x)} + g 3(x) ((6) ) g (x) + g (x) + g 3(x),, n N, f(x) n, ( ) (g (x) + g (x) + + g n (x)) g (x) + g (x) + + g n(x) (7) 6., g(x), f(x) g(x),, g(x + ) g(x) lim 0 6, n, (7)
, f(x), f(x + ) f(x) g(x + ) g(x) {g(x + ) g(x)} g(x + ) g(x) (8), (8), 0, ( ) (g(x)) g (x) (9), C R, (Cg(x)) Cg (x) (30),., (3x + 5x + ) (3x ) + (5x) + () ((7) ) 3(x ) + 5(x) + () ((30) ) 3 x + 5 + 0 (() ) 6x + 5, ( sin x + 5e x ) ( sin x) + (5e x ) ((7) ) (sin x) + 5(e x ) ((30) ) cos x + 5 e x ((8), () ) cos x + 5e x.3, g(x), (x), f(x) g(x)(x)
,, lim 0 g(x + ) g(x) (x + ) (x), lim 0 (3), (3), f(x),, f(x + ) f(x) g(x + )(x + ) g(x)(x) {g(x + ) g(x)} (x + ) + g(x) {(x + ) (x)} g(x + ) g(x) (x + ) (x) (x + ) + g(x) (3) 7, (3), 0, (g(x)(x)) g (x)(x) + g(x) (x) (33),, (x sin x) (x ) sin x + x (sin x) ((33) ) x sin x + x cos x ((), (8) ) x sin x + x cos x, f(x), f(x) g (x)g (x)g 3 (x),,,, (33), [g (x)g (x)g 3 (x)] [{g (x)g (x)}g 3 (x)] f(x) {g (x)g (x)}g 3 (x) {g (x)g (x)} g 3 (x) + {g (x)g (x)}g 3(x) ((33) ) {g (x)g (x) + g (x)g (x)}g 3 (x) + {g (x)g (x)}g 3(x) ((33) ) g (x)g (x)g 3 (x) + g (x)g (x)g 3 (x) + g (x)g (x)g 3(x) 7 (3),, g(x)(x + ). 3
,, n N, f(x) n, ( ) (g (x)g (x) g n (x)) g (x)g (x) g n (x) + g (x)g (x) g n (x) + + g (x)g (x) g n(x) (34) 8, g (x) g (x) g n (x) x, (34), (x n ) (x} x {{ x} n ) {(x) x x} + {x (x) x} + + {x x (x) } ((34) ) ( x x) + (x x) + + (x x ) nx n, (x),,, (34), ()..4, g(x), (x), f(x) g(x) (x), f(x), f(x) g(x) (x),.3, (33), ( ) g(x) ( ) g(x) (x) (x) g (x) (x) + g(x) ( ) (35) (x),, (x). 8, n, (34) 4
,, (x + ) (x) lim 0, (36), (x), { (x + ) } (x) (x + ) (x) (x + )(x) (x + ) (x) (37) (x + )(x), (37), 0, ( ) (36) ( ) (x) (x) ((x)) (38), n N, (x) x n,, (x n ) ( ) x n (xn ) (x n ) ((38) ) nxn x n nx n n ( n) x n (() ), (x n ) ( n) x n (39),, x n, x n,, x, () (39), m Z, (x m ) mx m (40) 5
,, (35), (38), ( ) g(x) ( ) g (x) (x) (x) + g(x) (x) ((35) ) g (x) (x) g(x) (x) ((x)) ((38) ) g (x)(x) g(x) (x) ((x)), ( ) ( ) g(x) g (x)(x) g(x) (x) (x) ((x)) (4),, ( ) x x (x) (x + ) x (x + ) + (x + ) ((4) ), (x + ) x x (x + ) ((), (6) ) x (x + ) (tan x) ( ) sin x cos x (sin x) cos x sin x (cos x) (cos x) cos x cos x sin x ( sin x) cos x cos x + sin x cos x cos x ((4) ) ((8) ).5, f(x) sin(x + ) f(x) e x sin x, f(x) sin(x + ), y x +, f(x), f(x) sin y, y x + 6
, g(y) sin y, (x) x +, f(x) e x sin x, y x sin x, f(x), f(x) e y, y x sin x, g(y) e y, (x) x sin x,, g(y), (x), f(x) g((x)) (4),, f(x), g(y) y y (x), f(x) (4) f(x) g(y) (x).,, lim y 0 g(y + y) g(y) (x + ) (x), lim y 0 (43), (43), f(x) 9, y (x),, f(x), f(x + ) f(x) g((x + )) g((x)) g((x + )) g(y) (44), (44), (43) g(y + y),,, (x + ) y + y (45), (45), y (x + ) y (x + ) (x) (46), y.,, (45), (44), (45), (46), f(x + ) f(x) g((x + )) g(y) 9, g(y), x, y ((44) ) 7
g(y + y) g(y) g(y + y) g(y) y g(y + y) g(y) y y (x + ) (x) ((45) ) ((46) ) (47), 0, y 0, (47), 0, d dg d (g((x))) (y) dx dy dx (x) (, y (x)) dg d ((x)) dy dx (x) 0, y (x), (I) f(x) y, y dg (y), y x dy, x d (x). dx (II) y (x), y x, (I) x., f(x) g((x))., f(x) sin(x + ), y x +, d dx sin(x + ) d dy sin y dy dx d cos y dx (x + ) cos(x + ) x (48) x cos(x + ), f(x) e x sin x, y x sin x, dg dy d dx ex sin x d dy ey dy dx e y d (x sin x) dx 0, g (y), (x), d (y), (x) dx 8
{ } d e x sin x dx (x) sin x + x d (sin x) dx (sin x + x cos x) e x sin x (49) 3, f : R R,, y R, f(x) y x R., y R, y f(x) x R, f,, f : R R.,,, x y, f ` (y), y R,,, x, x f (y) (50), (50) (5), f(x) y (5) f(f (y)) y (5), f (y),, (5) y 3., f(x) x, y < 0, y f(x) x R, y > 0, y f(x) x R, x ± y R., x y, x y,, x 0, y 0, f(x) x, f (y) y ( 4 )., n N, f(x) x n, x y f(x), x y,, x + x sin x y,, (48) (49), f(x) f (y),, y f(x) x 9
y x x y x y 0 x 0 y 4: x y,, x 0, y 0, f(x) x f (y) y., x 0, y 0, f(x) x n, f (y) y n.,,, (x) x n (53), (x), (x) x n, 0 x R, (x) x n, n x, (x) n x (54) 3,, (54) x, n(x) n (x) (55) 4, (53), (55), (x n ) (x) n(x) n ((55) ) n (x) n n (x n ) n ((53) ) n x n n n x n 3 x y, (54), (5). 4, y (x),.5, (54) 0
, ( ) (x n ) n x n (56),, x n, x,, x n, m Z, f(x) x m n, f(x), f(x) (x n ) m, (40), (56), (x m n ) [(x n ) m ] m(x n ) m (x n ) m(x n ) m n x n m n x m n + n ((40) ) ((56) ) m n x m n 5, ( ) (x m n ) m n x m n (57),, x m n m n., x,, x 3., f(x) e x, x R, e x > 0, x y f(x), y y > 0, f(x) e x,, f (y) log y ( 5 )., 3.,,, (x) log x (58) 5, y x n,.5,.
y e x x y x log y 0 x 0 y 5: y y > 0, f(x) e x f (y) log y., (x), (x) log x, 0 < x R, (x) log x, e x, e (x) x (59) 6,, (59) x, e (x) (x) (60) 7, (59), (60), (log x) (x) e (x) x ((60) ) ((59) ), (log x) (6) x, (6), f(x), 6 3., x y, (59), (5). 7, y (x),.5, (59)
log (log f(x)) f (x) (6) f(x) 8, (6),,, (59) (x) log x, 0 < x R, e log x x (63), α R, (63) α, x α, x α (e log x ) α e α log x (64), (64), (x α ) (e α log x ) ((64) ) e α log x (α log x) e α log x α (log x) x α α x αx α ((64), (6) ) 9, α R, (x α ) αx α (65),, x α, x α,, x 8, log y y > 0, (6) f(x) > 0, (6), f(x) f(x), {log( f(x))} f (x) f(x), f(x) < 0,, (log f(x) ) f (x) f(x),,,, (6),, 9, y α log x,.5,. 3
,, (64), x a, α x, 0 < a R x R, a x e x log a e (log a)x (66), λ log a, a x, ( ) e x, a x e λx, (66), (a x ) (e (log a)x ) ((66) ) e (log a)x ((log a)x) e (log a)x (log a) (x) a x (log a) ((66) ) (log a) a x 30, 0 < a R, ( ) (a x ) (log a) a x (67).4, a x, C R, (a x ) Ca x (68),, C y a x x 0, (67) (68), C, C log a, x, x R,, e x, e x +x e x e x (69)., y e x, y e x (70), (69), e x +x y y 30, y (log a)x,.5,. 4
,, (70), log(y y ) x + x (7) log y x, log y x, (7),, (69), x x, log(y y ) log y + log y (7) e x x e x e x ex e x,, ( ) y log log y log y (73) y, (7), (73), 0 < y, y R, ( ) log(y y ) log y + log y (74) ( ) y log log y log y (75) y,, (74) (75), (6) log,.3.4,, f(x) g(x)(x) (76), (76) log, (74), log f(x) log{g(x)(x)} log g(x) + log (x) (77) 3, (6), (77), f (x) f(x) g (x) g(x) + (x) (x) 3,, g(x) > 0, (x) > 0 5 (78)
,, (78) f(x), (g(x)(x)) f (x) { g } (x) f(x) g(x) + (x) (x) { g } (x) g(x)(x) g(x) + (x) (x) g (x)(x) + g(x) (x) ((78) ) ((76) ), f(x) g(x) (x), (79) log, (75), ( ) g(x) log f(x) log (x) (79) log g(x) log (x) (80) 3, (6), (80), f (x) f(x) g (x) g(x) (x) (x) (8), (8) f(x), ( ) g(x) f (x) (x) { g (x) f(x) g(x) (x) } g(x) (x) (x) { g } (x) g(x) (x) (x) g (x) (x) g(x) (x) (x) g (x)(x) g(x) (x) (x) ((8) ) ((79) ),.4 f(x) x x +,, (8) log, log f(x) log x log(x + ) (83) 3,, g(x) > 0, (x) > 0 (8) 6
, (83), f (x) f(x) (x) x (x + ) x + x x x + (x + ) x x(x + ) x x(x + ) (84), (84) f(x), ( ) x x f (x) + x f(x) x(x + ) x x + x x(x + ),, f(x) x (x + ) x + x, log, x log f(x) log + x ( ) x log + x ( ) x log + x {log( x ) log( + x ) },, f (x) f(x) { ( x ) x ( + x ) } + x { x x x } + x { ( x) x + } + x ( x) ( + x ) + ( x ) ( x )( + x ) 7
( x) ( x )( + x ) x ( x )( + x ) (85),, (85) f(x), ( ) x f x (x) + x ( x )( + x ) x + x ( x ) ( + x ) x + x {( x )( + x )} x ( + x ) x 4,,,,, log,, 3.3, f(x) sin x, cos x, tan x,, f(x) sin x, y y R, y f(x) x R, x y f(x), x y,, π x π, y, f(x) sin x, f (y) sin y ( 6 ). 33,,,, (x) sin x (86), (x), sin x, x x R, (x) sin x, sin x π (x) π (87) 33, sin y, sin y sin y arcsin y., x arcsin y, x sin x y,, arcsin y sin y (arc), arcsin y, sin,, sin y 8
y y sin x x π x sin y π 0 x π 0 y π 6: x y,, π x π, y, f(x) sin x f (y) sin y., sin (x) x (88) 34,, (88) x, {cos (x)} (x) (89) 35, (89), (sin x) (x) cos (x) (90), (87), (88), cos (x) sin (x) x, (90), sin` x (sin x) x, f(x) cos x, sin x, x y f(x), x y, 34 3., 3., x y, (88), (5). 35, y (x),.5, (88) 9
, 0 x π, y, f(x) cos x, f (y) cos y ( 7 ). 36 x π y 0 x π x cos y y cos x 0 y 7: x y,, 0 x π, y, f(x) cos x f (y) cos y.,,, (x) cos x, sin x, cos (x) x (9) 37 x, cos` x (cos x) x, ( cos y + π ) sin y, ( x cos y + π ) (9) sin y (93), (9), cos x y + π (94) 36 sin y, cos y arccos y 37 3., 3., x y, (9), (5). 30
, (93), y sin ( x), (94), cos x sin ( x) + π, cos x sin x,, sin x, cos x, f(x) tan x, sin x cos x, x y f(x), x π < x < π, f(x) tan x, f (y) tan y ( 8 ). 38 y tan x x π y 0 x π π 0 y x tan y π 8: x π < x < π, f(x) tan x f (y) tan y.,,, (x) tan x, (x), (tan x) cos x (95), 39 tan (x) x (96) 40 x, cos (x) (x) 38 sin y cos y, tan y arctan y 39 (95),.4. 40 3., 3., x y, (96), (5). 3
4, (tan x) (x), (96), x tan (x) cos (x) (97) sin (x) cos (x) cos (x) cos (x) cos (x), cos (x) + x, (97) tan` x (tan x) + x 4, y (x),.5, (96) 3