Derivades Infinit`esims (per x → 0) Fórmules de MacLaurin

Formulari
(funcions d’una variable)
Infinitèsims (per x → 0)
sin x = x + o(x)
1 − cos x =
1 2
x + o(x2 )
2
log(1 + x) = x + o(x)
Derivades
ax − 1 = x log a + o(x)
f0
f
(1 + x)α − 1 = αx + o(x)
xα
αxα−1 (α ∈ R)
tan x = x + o(x)
sin x
cos x
sinh x = x + o(x)
cos x
− sin x
loga x
1
x log a
ax
ax log a
tan x
cot x
arcsin x
arccos x
arctan x
arccot x
π
1
(x 6= + πn, n ∈ Z)
cos2 x
2
1
− 2 (x 6= πn, n ∈ Z)
sin x
1
√
(−1 < x < 1)
1 − x2
1
(−1 < x < 1)
−√
1 − x2
1
1 + x2
1
−
1 + x2
sinh x
cosh x
cosh x
sinh x
cosh x − 1 =
tanh x = x + o(x)
Fórmules de MacLaurin
ex =
n
X
xk
k=0
sin x =
n
X
k=1
cos x =
coth x
1
cosh2 x
1
−
sinh2 x
k!
(−1)k−1
n
X
log(1 + x) =
n
X
+ Rn+1 (x)
x2k−1
+ R2n+1 (x)
(2k − 1)!
(−1)k
k=0
tanh x
1 2
x + o(x2 )
2
x2k
+ R2n+2 (x)
(2k)!
(−1)k−1
k=1
(1 + x)α = 1 +
xk
+ Rn+1 (x)
k
n
X
α(α − 1) · · · (α − k + 1)
k=1
k!
xk + Rn+1 (x)
Primitives
R
f
0
C
1
x+C
xα
xα+1
+C
α+1
1
x
log |x| + C
ax
ax
+C
log a
sin x
− cos x + C
cos x
sin x + C
1
cos2 x
tan x + C
1
sin2 x
1
√
1 − x2
1
1 + x2
1
√
x2 ± 1
1
1 − x2
(x 6=
π
+ πn, n ∈ Z)
2
− cot x + C
(x 6= πn, n ∈ Z)
arcsin x + C
(−1 < x < 1)
arctan x + C
√
log |x + x2 ± 1| + C
1 + x
1
2 log 1 − x + C
sinh x
cosh x + C
cosh x
sinh x + C
1
cosh2 x
1
sinh2 x
f
tanh x + C
− coth x + C