Tabla de Derivadas

TABLA DERIVADAS
1
2
3
4
5
6
7
8
9
10
11
12
Por: Fernando Valdés M ©,
UTP, Pereira
d
(cu) = cu′
dx
d
(uvw) = u′ vw + uv ′ w + uvw′
dx
d n
(u ) = nun−1 u′
dx
d
(u + v) = u′ + v ′
dx
d ( u ) vu′ − uv ′
=
dx v
v2
d
u ′
|u| =
u
dx
|u|
d
(uv) = u′ v + uv ′
dx
d
(c) = 0
dx
d
1
ln(u) = u′
dx
u
d u
(e ) = eu u′
dx
d
cos(u) = − sen(u) u′
dx
d
sec(u) = sec(u) tan(u) u′
dx
d
f (g(x)) = f ′ (g(x))g ′ (x)
dx
d
sen(u) = cos(u) u′
dx
d
cos(ku) = − sen(ku)ku′
dx
d
cot(u) = − csc2 (u) u′
dx
1
d
loga (u) =
u′
dx
u ln(a)
d
sen(ku) = cos(ku) ku′
dx
d
tan(u) = sec2 (u) u′
dx
d
csc(u) = − csc(u) cot(u) u′
dx
d u
a = au ln(a) u′
dx
1
d
arc sen(u) = √
u′
dx
1 − u2
d
senh(u) = cosh(u) u′
dx
d
sech(u) = −sech(u) tanh(u) u′
dx
d
1
u′
senh−1 (u) = √
dx
1 + u2
d
1
sech−1 (u) = − √
u′
dx
u 1 − u2
1
d
arc cos(u) = − √
u′
dx
1 − u2
1
d
arctan(u) =
u′
dx
1 + u2
d
cosh(u) = senh(u) u′
dx
d
csch(u) = −csch(u) coth(u) u′
dx
d
1
u′
cosh−1 (u) = √
dx
u2 − 1
d
1
csch−1 (u) = − √
u′
dx
|u| 1 + u2
d
tanh(u) = sech2 (u) u′
dx
d
coth(u) = −csch2 (u) u′
dx
d
1
tanh−1 (u) =
u′
dx
1 − u2
TRIGONOMETRÍA HIPERBÓLICA
e +e
ex − e−x
senh(x)
cosh(x) =
senh(x) =
tanh(x) =
2
2
cosh(x)
2
2
2
2
2
cosh (x) − senh (x) = 1
1 − tanh (x) = sech (x)
coth (x) − 1 = csch2 (x)
(
)
√
√
1
1+x
senh−1 (x) = ln(x + x2 + 1) cosh−1 (x) = ln(x + x2 − 1) tanh−1 (x) = ln
2
1−x
−1
+
cosh(2x)
1
+
cosh(2x)
senh2 (x) =
cosh2 (x) =
cosh(2x) = cosh2 (x) + senh2 (x)
2
2
−x
x
1
2
3
4
2
2
TRIGONOMETRÍA CIRCULAR
1 + tan2 (x) = sec2 (x)
sin(π − x) = sen(x),
sen(2x) = 2 sen(x) cos(x)
1 − cos(2x)
sen2 (x) =
2
cos(−x) = cos(x)
sec(x) = 1/ cos(x)
5
6
sen (x) + cos (x) = 1
sen(x + π/2) = cos(x)
cos(x + π/2) = sen(x)
2 tan(x)
tan(2x) =
1 − tan2 (x)
sen(−x) = − sen(x),
csc(x) = 1/ sen(x),
7
tan(x) =
8
a sen(x) + b cos(x) = R sen(x + α)
9
a sen(x) + b cos(x) = R cos(x − α)
1
2
3
4
10
d
1
coth−1 (u) =
u′
dx
1 − u2
sen
−1
cos(x)
cot(x) =
sen(x)
√
R = a2 + b2 =⇒
√
R = a2 + b2
=⇒
sen(x)
cos(x)
(x) + cos
−1
=⇒
=⇒
tan−1 (x) + cot−1 (x) = π/2
(x) = π/2
1
1 + cot2 (x) = csc2 (x)
sen(π/2 − x) = cos(x)
cos(2x) = cos2 (x) − sen2 (x)
1 + cos(2x)
cos2 (x) =
2
tan(−x) = − tan(x)
cot(x) = 1/ tan(x)
b
a a
tan(α) =
b
tan−1 (x) + tan−1 (1/x) = π/2
tan(α) =