"
"
"
x - x0 =: x
x
x
f = f(x) - f(x0)
f
f f(x) - f(x0)
=
x x - x0
f
f x0
x x0
f f(x) - f(x0)
lim = lim .
xx0 xx0 - x0
x x
y = f(x)
dy df(x)
y , f (x), , , Ź
dx dx
x0 Ä…
x0 OX
f x0
f (x0) x0
 y = x
y = x2
y = x3
y = sin x
y = ex
(c) = 0 c " R
(xa) = axa-1 x > 0 a " R
(sin x) = cos x
(cos x) = - sin x
1
( x) = cos x = 0

cos2 x
-1
( x) = sin x = 0

sin2 x
1
Ä„
(arcsin x) = " -1 < x < 1 -Ä„ arcsin x
2 2
1 - x2
-1
(arccos x) = " -1 < x < 1 0 arccos x Ä„
1 - x2
1
Ä„
( x) = -Ä„ < x <
2
1 + x2 2
-1
( x) = 0 < x < Ä„
1 + x2
(ex) = ex
(ax) = ax ln x a > 0
1
(ln |x|) = x = 0

x
1
(loga |x|) = a > 0 a = 1 x = 0

x ln a
f g x
(c · f(x)) = c · f (x) c " R
(f(x) Ä… g(x)) = f (x) Ä… g (x)
(f(x) · g(x)) = f (x)g(x) + f(x)g (x)

f(x) f (x)g(x) - f(x)g (x)
= g(x) = 0

g(x)
[g(x)]2
"
2
f(x) = x5 + 4x3 - 6x2 - + x
x
g(x) = x2 ln x
ex
h(x) =
sin x
x2 - 1
s(x) =
x2 + 1
f x
g f(x)
[g(f(x))] = g (f(x)) · f (x).
f(x) = cos3 x
g(x) = (3x2 + 2x - 10)4
2
h(x) = esin x
1
s(x) = "
cos3 x3 - 1
2
f(x) = xx
g(x) = xsin x
m3
V = 10 d = 2mm
min
D = 1000m
R = 20cm
3
V = 100cm
sek
h
1
V = Ä„h2(3R-
3
h)
f (x0, f(x0))
y - f(x0) = f (x0)(x - x0).
 f(x) = ex (0, 1)
g(x) = sin x (Ä„, 0)
A = (2, -1) B = (4, 3)
y = x2 + 1 C ABC
1
y =
x + 1
S = 2
f x0 df x = x-x0
df( x) := f (x0) x.
f x0
f(x0 + x) H" f(x0) + f (x0) x.
f
f = f(x0 + x) - f(x0)
df = f (x0) x x
f - df
lim = 0.
x0
x
"
4
15.96
arctan 1.05
e-0.001
ln 1.004
V = 8cm3
0.3cm3
2m
3cm
 x y y = f(x)
f (x0) x0 x
x x
y y
y H" |f (x0)| x.
Ä…1mm
65mm
100m 0.01sek
10.00sek
y = f(x)

f (x) = f (x) ,
y = f(x)

f (x) = f (x) ,

f(4)(x) = f (x) ,

f(n)(x) = f(n-1)(x) .
y = f(x)
d2f(x) d3f(x)
y
, ÿ ,
dx2 dx3
f , f , f
2
f(x) = ex
g(x) = x ln x
h(x) = sin3 x