W6.pdf

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wyklady.dvi
1 Definicja
f : A →
R
A
⊆
R
lim
h→0
f (x 0 + h) − f (x 0 )
h
,
f
x 0
f ′ (x 0 )
h
f (x 0 + h) − f (x 0 )
h
f
x 0
+
h
2 Przykład
f (x) = 2 x 0 = 1
f (x 0 + h) − f (x 0 )
h
= f (1 + h) − f (1)
h
= 2 − 2
h
=
0
h = 0 −→ 0
h → 0.
f (x) = 2x
x 0 = −2
f (x 0 + h) − f (x 0 )
h
= f (−2 + h) − f (−2)
h
= 2(−2 + h) − 2(−2)
h
= 2(−2 + h + 2)
h
= 2h)
h
= 2 −→ 2
h
→ 0.
3 Definicja
I
x 0
4 T WIERDZENIE
5 Przykład
f (x) = |x|
x 0 = 0
8
<
f (x 0 + h) − f (x 0 )
h
= f (h) − f (0)
h
|h| − 0
h
|h|
h
h
h = 1,
h > 0,
=
=
=
:
−h
h
= −1,
h < 0,
6 Definicja
f
f ′ : x → f ′ (x)
f
+
47713884.030.png 47713884.031.png 47713884.032.png 47713884.033.png 47713884.001.png 47713884.002.png 47713884.003.png
+
+
df
dx
+
(c) ′ = 0,
(x) ′ = 1,
(x n ) ′ = n x n−1 ,
(e x ) ′ = e x ,
(a x ) ′ = a x ln a,
a
∈
R + \ {1},
(ln |x|) ′ =
1
x ,
x ∈
R + ,
1
x ln a ,
(log a x) ′ =
a
∈
R + \ {1},
(sin x) ′ = cos x,
(cos x) ′ = − sin x,
( tg x) ′ =
cos 2 x = 1 + tg 2 x,
1
x
=
2 + k, k
∈
Z,
( ctg x) ′ = − 1
sin 2 x = −(1 + ctg 2 x),
x
= + k, k
∈
Z,
1
(arcsin x) ′ =
√
1 − x 2 ,
x ∈ (−1, 1),
−1
(arccos x) ′ =
√
1 − x 2 ,
x ∈ (−1, 1),
( arctg x) ′ = 1
1 + x 2 ,
( arcctg x) ′ = − 1
1 + x 2 ,
(sinh x) ′ = cosh x,
(cosh x) ′ = sinh x,
( tgh x) ′ =
1
cosh 2 x ,
−1
sinh 2 x .
( ctgh x) ′ =
7 T WIERDZENIE
f
g
47713884.004.png 47713884.005.png 47713884.006.png 47713884.007.png
(f
± g) ′ (x) = f ′ (x) ± g ′ (x)
(f
g) ′ (x) = f ′ (x) g(x) + f (x) g ′ (x)
x
g(x) = 0
f
g
′
(x) = f ′ (x) g(x) − f (x) g ′ (x)
[g(x)] 2
.
8 Przykład
c
f
[(cf )(x)] ′ = c ′ f (x) + c f ′ (x) = 0 f (x) + c f ′ (x) = c f ′ (x).
f (x) = 0
1
f (x)
′
= 1 ′ f (x) − 1 f ′ (x)
[f (x)] 2
−f ′ (x)
[f (x)] 2 .
=
(a n x n +a n−1 x n−1 + +a 2 x 2 +a 1 x+a 0 ) ′ = na n x n−1 +(n−1)a n−1 x n−2 + +2a 2 x+a 1 .
ln x e x
2x 2
′
= (ln x e x ) ′ 2x 2 − (ln x e x ) (2x 2 ) ′
[2x 2 ] 2
= [(ln x) ′ e x + ln x (e x ) ′ ] 2x 2 − (ln x e x ) (2 2x)
4x 4
2x 2 − 4x ln x e x
4x 4
= e x 2x + 2x x ln x − 4x ln x
4x 4
1
x
e x + ln x e x
= e x 1 + x ln x − 2 ln x
2x 3
.
( tg x) ′ =
sin x
cos x
= sin ′ x cos x − sin x cos ′ x
cos 2 x
= cos x cos x + sin x sin x
cos 2 x
=
1
cos 2 x
9 T WIERDZENIE
x 0
g
x 0
f ◦g
f
g 0 = g(x 0 )
f
◦ g
x 0
(f
◦ g) ′ (x 0 ) = f ′ (g(x 0 )) g ′ (x 0 ) = f ′ (g 0 ) g ′ (x 0 ).
=
47713884.008.png 47713884.009.png 47713884.010.png 47713884.011.png 47713884.012.png
+
d(f ◦ g)
dx
(x 0 ) =
df
dg
(g 0 ) dg
dx
(x 0 ).
10 Przykład
e 1/x
′ = e 1/x −1
x 2
=
−e 1/x
x 2
(x x ) ′ =
e lnx
x
=
e xln x
′ = e xln x (xln x) ′ = x x
1 ln x + x 1
x
= x x (ln x+1)
(sin x) tg x
′ = e ln sin x tg x (ln sin x tg x) ′
= (sin x) tg x [(ln sin x) ′ tg x + ln sin x ( tg x) ′ ]
e ln sin x tg x
= (sin x) tg x
1
sin x
(sin x) ′ tg x + ln sin x 1
cos 2 x
= (sin x) tg x
cos x
sin x
tg x + ln sin x
cos 2 x
1 + ln sin x
cos 2 x
= (sin x) tg x
sin(x tg x ) ′ = cos(x tg x ) (x tg x ) ′ = cos(x tg x )
(e ln x ) tg x
′ = cos(x tg x ) (e ln x tg x ) ′
= cos(x tg x ) e ln x tg x (ln x tg x) ′ = cos(x tg x ) x tg x
tg x
x
+
ln x
cos 2 x
11 T WIERDZENIE
f
f −1
f −1 (x)
′ =
1
f ′ (f −1 (x)) .
+
dy
dx =
1
dx
dy
.
12 Przykład
(ln x) ′ =
1
(e y ) ′
=
1
e y
=
1
e ln x =
1
x
y=ln x
y=ln x
′ =
47713884.013.png 47713884.014.png 47713884.015.png 47713884.016.png 47713884.017.png 47713884.018.png 47713884.019.png 47713884.020.png 47713884.021.png 47713884.022.png
(arccos x) ′ =
1
(cos y) ′
=
1
− sin y
y=arccos x
y=arccos x
(∗)
=
−1
=
√
−1
1 − cos 2 y
1 − x 2
y=arccos x
(∗)
f
(0, )
x 0
f
x 0
f ′′ (x 0 )
x
f
x →
f ′′ (x),
+
14 Przykład
f (x) = −4x 3 + 2x,
f ′ (x) = −12x 2 + 2,
f ′′ (x) = −24x,
f ′′′ (x) = −24,
f (IV ) (x) = 0.
+
f (x) = cos ln x,
x ∈
R + ,
f ′ (x) = − sin ln x 1
x =
− sin ln x
x
,
f ′′ (x) =
− cos ln x x
x − (− sin ln x) 1
x 2
= sin ln x − cos ln x
x 2
−1
x
=
[f (x) + f ′ (x)].
13 Definicja
47713884.023.png 47713884.024.png 47713884.025.png 47713884.026.png 47713884.027.png 47713884.028.png 47713884.029.png

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