f : (a, b) !

R x0 2 (a, b)

f (x0 + h) f (x0 )
f 0 (x0 ) := lim .
h!0 h
f 0 (x0 )
f (x0 , f (x0 ))

y = f 0 (x0 )(x x0 ) + f (x0 ) .

f 0 (x0 )
(x0 , f (x0 ))
0
f (x0 ) f
x0
f (x0 + h) = f (x0 ) + f 0 (x0 ) · h + r(h) ,
| {z } |{z}

r(h)
r(h) h limh!0 h
=0

x=0
( (
x
1+ex10
x 6= 0 4x2 2x x>0
f (x) = , g(x) = .
0 x=0 e 2x 1 x0

2
p sin x cos x ex ln(x + 1)
3
x+2 x5
↓ f g
( (
x sin x1 x 6= 0 x2 sin x1 x 6= 0
f (x) = g(x) =
0 x=0 0 x=0

x=0
a, b 2 R
(
5x2 + ax x0
f (x) =
2x + b x>0
 •

(f (x) ± g(x))0 = f 0 (x) ± g 0 (x); (c · f (x))0 = c · f 0 (x);

✓ ◆0
f (x) f 0 (x) · g(x) f (x) · g 0 (x)
(f (x) · g(x))0 = f 0 (x) · g(x) + f (x) · g 0 (x); = .
g(x) g 2 (x)

• [f (g(x))]0 = f 0 (g(x)) · g 0 (x)

•

1 0 1
(f ) (y) = , y = f (x)
f 0 (x)

•
1
(xa )0 = a · xa 1
; (ex )0 = ex ln0 (x) =
x
1
sin0 (x) = cos(x); cos0 (x) = sin(x); tg0 (x) = = 1 + tg2 (x)
cos2 (x)
1 1 1
arctg0 (x) = ; arcsin0 (x) = p ; arccos0 (x) = p .
1 + x2 1 x2 1 x2

2 1 1
tg x ctg x xex x ln x f (x) f 2 (x)
p
cos3 (4x) ln x xx (sin x)cos x
⇣ ⌘

·
x2 · ln 1+x
1 x xsin x
p
f (x) = x arctg x + 1 x2
x2 ln x (1, 0)

f : (a, b) ! (c, d)
(a, b) x0 2 (a, b) f 0 (x0 ) 6= 0 f 1 : (c, d) ! (a, b)
y0 = f (x0 )

1 0 1
(f ) (y0 ) = .
f 0 (x0 )

arctg(x)
arcsin(x) arccos(x)

✓ ◆
1 x
arctg x + arctg .
1+x
 ✓ ◆
1 x ⇡
arctg x + arctg = .
1+x 4

(
e 1/x2 x 6= 0
f (x) =
0 x=0

x=0
f x

f (x + h) f (x h) 0
lim = f (x).
h!0 2h

g(x) = ln pxx+a
2 +b2
+ a
b arctg xb .
ex e x
sinh(x) := 2
ex +e x sinh(x)
cosh(x) := 2 th(x) = cosh(x)

✓ ◆
1 1+x
arcth x = ln .
2 1 x

f :R!R f (2x) = 2f (x)

x2R x = 0 holokwirm
 f : (a, b) !R

• f x0 2 (a, b)
f 0 (x0 ) = 0
f f f 0 (x)

• f [c, d] ⇢
(a, b) (a, b)
x 2 (c, d) f 0 (x) > 0 f [c, d]
0
x 2 (c, d) f (x) 0 f [c, d]
0
x 2 (c, d) f (x) < 0 f [c, d]
x 2 (c, d) f 0 (x)  0 f [c, d]

ln x
x

↑⇣ ⌘ f (x) = xe x2

p1 , +1
2
A x>0 ex > 1 + x sin x < x
x2
cos x > 1 2
x 2 (0, ⇡2 )

sin x + tg x 2x.

ex  (1 + x)(1+x) x 0

↑ p
a, b, x 2 R

a sin x + b cos x  a 2 + b2 .

A
ex = x2 + 2x + 3
 1
A = {en : n 2 N}.
n2 + 3n 6

f : (a, b) ! R
0
f x 2 (a, b) f+ (x)
f 0 (x) f
(a, b)
f (a, b) f 0 (x) (a, b) f
(a, b)
f (a, b) x 2 (a, b)
f 00 (x) 0 f (a, b)

sin(x) (0, ⇡)

p x ⇡
3
x (0, +1), tg2 (0, ), ln x (0, +1).
2 2

x y a b
a+b=1

xa y b  ax + by.

Pn f :A!R
t1 , . . . , t n i=1 ti =1
x1 , . . . , x n 2 A
!
X X
f ti · x i  ti · f (xi ) .
i i
 n 1
p p p np 2
12 + 1 + 22 + 1 + . . . + n 2 + 1 n + 2n + 5 .
2
1
W q q
3 p
3 3 p
3
p
3
3+ 3+ 3 3  2 3.

↑ ↵

↵ 1
tg2 + tg2 + tg2 .
2 2 2 3

✓ ◆a+b+c
a 2 + b2 + c 2
a a bb c c ,
a+b+c
a b c
* a, b, c, d 2 R
q
a+b+c+d
1+e 4  4 (1 + ea )(1 + eb )(1 + ec )(1 + ed ) .

↑ n
v = 1

•
•
•
•
•
•
•

x2 ln x
p>0 f (x) = xp e x x 0
 f :R!R f 0 (0) = 1
">0 f ( ", ")
x y z

x2 y2 z2 3
xe + ye + ze p .
2e

e⇡ ⇡e
p
fn (x) = n exp(x) n2N fn
[0, 1]

P Y ✓ 1 + x k ◆p k
1 + k pk x k
P  ,
1 k pk x k 1 xk
k
P
0 < xk < 1 k pk = 1
x, y, z
1 1 1
+ + 6.
sin( x2 ) sin( y2 ) sin( z2 )
 f : [a, b] ! R
(a, b) c 2 (a, b)
-

f (a) f (b)
= f 0 (c) .
a b
f (a) f (b)

f : [a, b] ! R (a, b)
f (a) = f (b) c 2 (a, b)

f 0 (c) = 0 .

W (x) n
0
W (x) n 1

lim arctg(n + 1) arctg n.

n!1

lim x2 [arctg(x + (x)) arctg(x + (x))] ,

x!1
p
(x) = (1 + 1/x)x (x) = x
x
x y>0
2(x y) 2(x y)
 ln x2 ln y 2  .
x y

⇡
0 <↵ 2

↵ ↵
 tg ↵ tg  .
cos2 cos2 ↵

f, g : [0, 1] ! R (0, 1)
f (0) = f (1) = 0

g 0 (x)f (x) + f 0 (x) = 0

 (0, 1)
T f : R ! R
f (0) = f 0 (0) = f (1) = 0 x 2 (0, 1) f 00 (x) = 0
↑ f : [a, b] ! R (a, b)
b a ⇡ x0 f 0 (x 0) <1+ f 2 (x 0)

f, g : R ! R a2R

xf (a) af (x) g(x)f (a) g(a)f (x)

(a) lim (b) lim
x!a x a x!a x a

f :R!R R \ {0}
g1 = limx!0 f 0 (x) g2 = limx!0+ f 0 (x) f
0 f 0 (0) = g1
f, g : [a, b] ! R (a, b)
f (a) g(a) f 0 (x) g 0 (x)
f (b) = g(b) x 2 (a, b) f (x) = g(x)
f : [0, 1) ! R (0, 1)
f 0 (x) f (x) x>0 f (0) = 0
f (x) 0 x 0
f : [a, b] ! R (a, b)
|f 0 (x)|  C|f (x)| f (a) = 0 f ⌘0
p b a a+b
a, b > 0 a 6= b ab < ln b ln a < 2
 f : (a, b) ! R n 1
(a, b) x0 2 (a, b) f (n) (x0 ) x0
f
n
X f (k) (x0 )
f (x) = (x x0 )k + Rn (x, x0 ).
k!
k=0

n
Rn (x) = o(x x0 ) f (n + 1)
(a, b)
f (n+1) (✓)
• Rn (x, x0 ) = (n+1)!
(x x0 )n+1 ✓ 2 (x0 , x)
(n+1)
f (✓)
• Rn (x, x0 ) = n!
(x ✓)n (x x0 ) ✓ 2 (x0 , x)
(n+1)
f (✓)
• Rn (x, x0 ) = n!p
(x ✓)n+1 p
(x x0 ) p ✓ 2 (x0 , x) p>0

x=1 W (x) = x3 + 2x2 + 5

x=0 tg x

sin(sin x) x=0
5
2
n ex
n sin2 x
A 1
2+x
1
(x+1)2
x=0
n arctg x x2R
arctg x n

p
A 1+x x=0
1
(1 + x) 2
p1 10 4
2
p
3
 0 1
0 1

f, g : (a, b) ! R
g 0 (x) 6= 0 x b
limx!b f (x) = limx!b g(x) = 0 limx!b g(x) = ±1

f 0 (x)
C = lim ,
x!b g 0 (x)

f (x)
lim =C .
x!b g(x)

1 2 1 2
ex 1 x ex 1 x 2x esin x 1 x 2x
lim , lim , lim .
x!0 x2 x!0 x3 x!0 x3

x sin x arctg x
lim , lim .
x!0 x tg x x!0 ln(1 + x)

~ ✓pd ◆1/x2 .

1 1 sin x
lim , lim .
x!0 x ex 1 x!0 x

sin(sin x) sin x
lim .
x!0 sin(tg x) tg x

✓ ◆1/x
ln(1 + x)
lim .
x!0 x
 f : R ! R C1
|f (n) (x)|  C C>0
1
X f (k) (x0 )
f (x) = (x x0 ) k , x0 , x 2 R
k!
k=0

f :R!R f 00 (x0 )

f (x0 + h) 2f (x0 ) + f (x0 h)

f 00 (x0 ) = lim .
h!0 h2

f : R ! R |f 00 |  1
limx!1 f (x) = 0 limx!1 f 0 (x) =0

tg(sin(x)) sin(tg(x))
lim .
x!0 x7
 fn : A ! R
A⇢R f :A!R fn
• f A fn !A f x2A

lim fn (x) = f (x) ;

n!+1

• f A fn ◆A f

lim sup |fn (x) f (x)| = 0 ;

n!+1 x2A

• f A K⇢A fn ◆K f

fn : (a, b) ! R
f : (a, b) ! R f

fn (x) = nxn (1 x) x 2 [0, 1]

xn gn
limn!1 fn (xn )
fn : [0, 1] ! R
maxx2[0,1] fn (x)
A fn (x) = xn (1 x)
[0, 1]
p
# fn (x) = n
1 + xn
x 0
1
fn (x) = 1+nx
x 0

fn : [a, b] ! R
f : [a, b] ! R
f1  f2  f3 . . . .
fn ◆ f
fn : [a, b] ! R
f : [a, b] ! R fn fn ◆ f
 ⇣ ⌘
fn (x) = exp x + p1 n = 2, 3, . . .
ln n
[0, a] a>0
f :R!R fn (x) := f ( nx )
[0, a]
↑ fn (x) = n sin nx
R [0, a] a>0
fn ◆A f gn ◆A g

fn + gn ◆A f + g
fn · gn ◆A f · g
f g

P
fn (x)
x2
[ 1, 1] fn (x) = (1+x2 )n 1

P fn : A ! R
A⇢R n fn
P
Sk (x) := kn=1 fn (x)

P P
n supx2A |fn | fn
A

P
fn : [0, 1] ! R P n fn
n supx2[0,1] |fn | =1

X
(x ln x)n
n

(0, 1]
A
X sin(n3 x)

n
n2
↑
Xp
n4 x
f (x) = xe
n

[0, +1)
P p n2 x
f (x) = n xe
(0, +1) f (x)

P1 2 x2 e n2 |x|
n=0 n R

1-
↓
1 P1
W n=0 x2 (1 x2 ) n ( 1, 1)
P1 n sin 1
n=0 2 3n x (0, 1)
+
↑ P1
n=0 |x|
p e nx2 p>0 x2R We &

⇣ ⌘
A P1
n=2 ln 1+ x2
n ln2 n
x2R
P1 p
n
n=0 |x| x 2 ( 1, 1)
P1 ⇣ ⌘ ⇣
⌘
x2
n=1 exp n 2 1 x2R
P1 ⇣ ⌘
e n
n=1 sin x x 2 (0, 1)

Pn f n , gn : A ! R
Gn (x) = k=1 gk (x)
|Gn (x)|  C x2A n
P1
n=1 |fn (x) fn+1 (x)|
fn ! 0
P1 P1 P1
n=1 f n gn n=1 f n gn = n=1 Gn (fn fn+1 ).

Pn f n , gn : A ! R
Gn (x) = k=1 gk (x)

|Gn (x)|  C x2A n

fn ! 0
P1 P1 P1
n=1 f n gn n=1 f n gn = n=1 Gn (fn fn+1 ).
P1 ( 1)n+1
n=1 n+x2 x2R
P1 sin nx
n=1 n x 2 (0, 2⇡)
P1 ( 1)n+1
n=1 n+x2
arctan(nx) x 2 R

fn R fn ◆R f f
R
P
P fn 2 C([0, 1]) n fn (x) [0, 1)
n fn (1)
P1
P1 n=1 fn (x)
[0, 1] |f
n=1 n (x)| [0, 1]
fn A f x0
A n limx!x0 fn (x)

lim lim fn (x) = lim f (x),

n!1 x!x0 x!x0

(Pn ) R
f f
C>0 n 1
n
X sin kx
 C.
k
k=1
 (fn ) [a, b]
(a, b) x0 2 [a, b] (fn (x0 ))
fn0 (a, b) g
(fn ) f (a, b)
f0 = g

A 1
X
nan ,
n=1
|a| < 1

1
X 1
.
n2n
n=1

A
X x2
f (x) =
n
n4 + x 4

P1
↓ 1 2
C1 (R)
P n=1x x +n
2 2

A f (x) = n n2 +x2

P
n kfn k
P n4 x
f (x) = ne (0, +1)
P1 1
S(x) = n=1 n2 ln(1 + n2 x2 )
S(x)
R
S(x) R \ {0}
S 0 (0)
S C1 R \ {0}
 Fn (x)
8
>
<xn
3 x 2 [0, n1 ]
0
Fn (0) = 0 Fn (x) = 2n2 xn3 x 2 [ n1 , n2 ]
>
:
0 x

• Fn x=0
0
• Fn

• Fn (x) x>0
P1 1 x
f (x) = n=1 n sin n C1 f

f :R!R P C 1 (R) f (0) = f 0 (0) =

0 F (x) = 1
n=1 f
x
n

C1
 1
X
f (x) = an · xn .
n=0

P1
n=0 an xn
1
R := p 2 [0, +1]
lim supn!+1 n |an |

•
x 2 ( R, R)
• |x| > R
P1
an xn
p
n
n=0

P1 lim supn!1 |an |

n=0 a n xn x R
|x| < R |x| > R R

bn bn1 , b n2 , . . . , b nk
g1 , g 2 , . . . , g k bk

lim sup bn = max{g1 , g2 , . . . , gk }.

n!1

1
X 1
X
xn 1
(a) = exp x, (b) ( 1)n+1 xn = ln(1 + x),
n! n
n=0 n=1
X1 X1
2n (2 + ( 1)n )n n
(c) 2
x2n , (d) x .
n n5 + 1
n=1 n=1
 1
X x2n+1
(a) ( 1)n = arctg x,
2n + 1
n=0
X1
(b) (3 + 2( 1)n ))n x3n ,
n=1
X1
n
(c) (ln n) 8 xn .
n=1

1
X 1
X
xn nn
xn .
n n!
n=1 n=1

1 ✓ ◆
X
p p
(1 + x) = xn ,
n
n=1

/N
p2
P1 n
P1 n
n=0 an x n=0 bn x
P1 R1 R2 R1 6= R2 R
n
n=0 (an +bn )x R = min(R1 , R2 ) R1 = R 2 R R1 = R2

sin(x) =
P1 n x2n+1 P1 n 1 xn
n=0 ( 1) (2n+1)! x2R ln(1 + x) = n=1 ( 1) n
x 2 ( 1, +1)

( 1
e x2 x 6= 0
f (x) =
0 x = 0.

C1 f (n) (0) = 0 n2N

f

f : A ! R x0 2 A
R>0
X1
f (n) (x0 )
f (x) = · (x x0 )n , |x x0 | < R.
n=0
n!
 f, g : R ! R x0 f ·g
x0
P
f (x) = 1n=0 an x
n R>
P1 f (n) (x0 )
0 x0 2 ( R, R) f (x) = n=0 n! (x x0 )n |x x0 | < R |x0 |

1
X
an xn ,
n=0

f (x) x 2 ( R, R)
1
X
n · an xn 1
,
n=0

0
f (x) ( R, R)

0
f (x) = xf (x).

P1 x3n
f :R!R f (x) = n=0 (3n)!
f 00 (x) + f 0 (x) + f (x) = ex x2R

P1
f (x) = n=0 an · xn R |x| < R
|x| > R
|x| = R

P1
n=0 an · z n z
|z| = R
1
X
an · z n = lim f (t · z) .
t!1
n=0

1
X ( 1)n 1
ln(2) = .
n
n=1
 1
⇡ X ( 1)n
= .
4 2n + 1
n=0

+1
X 1
( 1)n .
n(n + 1)
n=1

P1 n
an 0 f (x) = n=0 an x P1
R=1 limx!1 f (x) n=0 an <1

f :R!R x0 f (x0 ) 6= 0
1
f x0
P1 P
f (x) = n=0 an x
n g(x) = 1 n=0 bn x
n

R > 0 xn ! 0 xn 6= 0
f (xn ) = g(xn ) a n = bn n 0 f =g
P1 n
f (x) = n=0 an x
R=1 an ! 0 n ! +1 limx!1 (1 x)f (x) = 0
|x|  1
1
X
1 (2n 3)!!
|x| = 1 (1 x2 ) (1 x2 ) n .
2 (2n)!!
n=2
p
|x| = 1 (1 x2 )
 f : I ! R I ⇢ R f
F : I !R R F 0 (x) = f (x) x2I
f f (x) dx
I
Z
f (x) dx = F (x) + C ,

Z Z
1 1
xa dx = xa+1 + C, a 6= 1 dx = ln |x| + C,
a+1 x
Z Z
ex dx = ex + C, cos xdx = sin x + C,
Z Z
1
sin x dx = cos x + C, dx = tg x + C,
cos2 x
Z Z
1 1
p dx = arcsin x + C, dx = arctg x + C .
1 x2 1 + x2

Z Z
f (x)g 0 (x) dx = f (x)g(x) f 0 (x)g(x) dx
Z
f 0 (g(x))g 0 (x) dx = f (g(x)) + C

Z Z
F (g(x))g 0 (x) dx = y = g(x), dy = g 0 (x) dx = F (y) dy
 Z Z
2x
(a) e dx, (b) tg(x) dx
Z Z
(c) xex dx, (d) ex sin(x) dx,
Z Z
(e) x arctg(x) dx, (f ) ln(x) dx
Z Z
2
(g) x3 ex dx, (h) x(arctan(x))2 dx,
Z Z
2
(i) x ln(x) dx, (j) sin(cos(x)) sin(x)dx,
Z Z
(k) sin(sin(cos(x))) cos(cos(x)) sin(x) dx, (l) cos(ln x) dx,
Z Z ✓ ◆
1 1
(m) x2 ex sin(x) dx, (n) tg dx,
x2 x
Z p
Z
x sin x
(o) e dx, (p) dx,
cos3 x
Z Z
p 1
(q) x3 3 x + 1 dx, (r) dx,
sin x
Z
(s) sin3 (x) dx .

f (x) = ex · W (x),

W 2 R[x] k
k k
f :I !R I f
f
(
sin x1 x 6= 0
f (x) =
0 x=0
f (0) = 1
f :R!R g :R!R
C1 f ·g
f : R ! R limx!0+ f (x) = 1 f

f :R!R
inf x2R |f (x)| = 0
 P (x)
f (x) = Q(x)
P, Q 2 R[x]

c
(ax+b)n
Dx+E
(Ax2 +Bx+C)n

Z Z
x x
(a) dx, (b) dx
(x2 + 1)3 (x2 + x + 1)2
Z Z
1 1
(c) dx, (d) dx .
(x + x + 1)3
2 (x + 1)2
2

Z Z
x2 1
(a) dx, (b) p dx .
1 + x8 1 + e2x
Z Z
1 x2
(c) 3
dx, (d) dx,
x +1 x2 4
Z 5
x + x4 8
(e) dx .
x3 4x

P (x,y)
R(x, y) x y R(x, y) = Q(x,y)
P, Q 2
R[x, y]

R
R(sin x, cos x) dx t = tg(x/2) s=
sin x c = cos x
R p
R(x, ax2 + bx + c) dx ax2 +bx+c
R p
y = ↵x+ R1 (y, y 2 ± 1) dy
p p
y2 + 1 t = tg x y2 1 z = cosh(x)
R p 2
R(x, ax2 + bx + c) dx ax + bx + c = (Ax + B)(Cx + D)
R p R q
Ax+B
R(x, (AX + B)(Cx + d))) dx = R(x, (Cx + D) Cx+D ) dx =
R q q
Ax+B Ax+B
R1 (x, Cx+D
) dx z = Cx+D
 Z p Z
1 + x2 1
(a) dx, (b) dx .
x 2 + cos x
Z Z
1
(c) tg5 x dx, (d) dx,
sin3 x
Z Z r
sin4 x x a
(e) dx, (f ) dx,
cos2 x x+a
Z Z
3 x 1
(g) p dx, (h) p dx,
5 x 2 4x x 1 + x + x2
Z Z p
1
(i) p dx, (j) (2 x) 2x2 + 3x 8 dx .
x 2 2x + 5