INFORMATYKA I rok 2 semestr
Lista wzorów
Pochodne
1. c = 0
2. (xn) = nxn-1 w szczególności:
" (x) = 1
a -a
" (x) =
x2
"
1
"
" ( x) =
2 x
3. (ax) = ax ln a w szególności: (ex) = ex
1 1
4. (loga x) = w szczególności: (ln x) =
x ln a x
5. (sin x) = cos x
6. (cos x) = - sin x
1
7. (tg x) =
cos2 x
-1
8. (ctg x) =
sin2 x
1
"
9. (arc sin x) =
1-x2
-1
"
10. (arc cos x) =
1-x2
1
11. (arc tg x) =
x2+1
-1
12. (arc cot x) =
x2+1
Własności:
1. (c · f(x)) = c · f (x)
2. (f(x) Ä… g(x)) = f (x) Ä… g (x)
3. (f(x) · g(x)) = f (x) · g(x) + f(x) · g (x)
f (x)·g(x)-f(x)·g (x)
4. (f(x)) =
g(x) g2(x)
5. (f(g(x))) = f (g(x)) · g (x)
Wzory:
"
m
n
n
" xm = x
1
" x-n =
xn
" f(x)g(x) = eg(x)·ln |f(x)|
1
Całki

xÄ…+1
1. xądx = + C, ą = -1, a w szczególności:

Ä…+1

" dx = x + C
"
dx
" "
= 2 x + C
x

dx 1
" = - + C
x2 x

dx
2. = ln |x| + C
x

ax
3. axdx = + C, a > 0, a = 1, a w szczególności

ln a

" exdx = ex + C

4. cos xdx = sin x + C

5. sin xdx = - cos x + C

dx
6. = tg x + C
cos2 x

dx
7. = - ctg x + C
sin2 x

dx
"
8. = arc sin x + C
1-x2

"-dx
9. = arc cos x + C
1-x2

dx
10. = arc tg x + C
1+x2

-dx
11. = arc ctg x + C
1+x2
Szareg Taylora
"

xn
1. ex =
n!
n=0
"

(-1)n
2. sin x = x2n+1
(2n+1)!
n=0
"

(-1)n
3. cos x = x2n
(2n)!
n=0
"

(-1)n+1
4. ln(1 + x) = xn
n
n=1
2