¡Descarga Ejemplos de derivadas. y más Apuntes en PDF de Cálculo solo en Docsity!
Apuntes del 12 de Marzo del 2019 Ejemplos de la derivada de la potencia de funciones trigonométricas 1.- d dx
( se n
n
u ) =
d dx ( sin u ) n = d dx v n = n v n − 1 dv dx = n ( sin u ) n − 1 cos u du dx a) d dx
( sin
10 2 x 3
d dx
( sin 2 x
3
10 = 10 ∗sin 9 2 x 3 ∗cos 2 x 3 ∗ 6 x 2 = 60 x 2 ∗sin 9 2 x 3 ∗cos 2 x 3 b) d dx
( sin
− 15 4 x 9
d dx
( sin 4 x
9
− 15 =− 15 ∗sin − 16 4 x 9 ∗cos 4 x 9 ∗ 36 x 8 =− 540 x 8 ∗sin − 16 4 x 9 ∗cos 4 x 9 c) d dx
(sin
4
5 − 3 x^5 )= d
dx
( sin− 3 x^5 )
4 5 =
sin − 1 (^5) − 3 x^5 ∗cos− 3 x^5 ∗− 15 x^4 =− 12 x^4 ∗sin − 1 (^5) − 3 x^5 ∗cos− 3 x^5 d) d dx
(sin
− 3
7 2 x^10 )= d
dx
( sin 2 x
10
− 3 7 =
sin − 10 (^7 2) x^10 ∗cos 2 x^10 ∗ 20 x^9 =−^60 7 x 9 ∗sin − 10 (^7 2) x^10 ∗cos 2 x^10 2.- d dx
( cos
n
u )=
d dx ( cos u ) n = n cos n − 1 u ∗−sin u du dx =− n cos n − 1 u ∗sin u du dx a) d dx
( cos
5 2 x 5
d dx
( cos 2 x
5
5 =− 5 ∗cos 4 2 x 5 ∗sin 2 x 5 ∗ 10 x 4 =− 50 x 4 ∗cos 4 2 x 5 ∗sin 2 x 5 b) d dx
( cos
− 7 2 x 5
d dx
( cos 2 x
5
− 7 = 7 ∗cos − 8 2 x 5 ∗sin 2 x 5 ∗ 10 x 4 = 70 x 4 ∗cos − 8 2 x 5 ∗sin 2 x 5 c) d dx
(cos
1
3 2 x^5 )= d
dx
( cos 2 x^5 )
1 3 =
∗cos − 2 (^3 2) x^5 ∗sin 2 x^5 ∗ 10 x^4 =−^10 3 x 4 ∗cos − 2 (^3 2) x^5 ∗sin 2 x^5 d) d dx
(cos
− 10
3 2 x^5 )= d
dx
( cos 2 x
5
− 10 3 =
∗cos − 13 (^3 2) x^5 ∗sin 2 x^5 ∗ 10 x^4 =^100 3 x 4 ∗cos − 13 (^3 2) x^5 ∗sin 2 x^5 3.- d dx
( tan
n
u )=
d dx ( tan u ) n = n tan n − 1 u ∗ sec 2 u du dx a) d dx
( tan
8 3 x 4
d dx
( tan 3 x
4
8 = 8 ∗tan 7 3 x 4 ∗ sec 2 3 x 4 ∗ 12 x 3 = 96 x 3 ∗tan 7 3 x 4 ∗ sec 2 3 x 4 b) d dx
( tan
− 7 − 2 x 5
d dx
( tan − 2 x
5
− 7 =− 7 ∗tan − 8 − 2 x 5 ∗ sec 2 − 2 x 5 ∗− 10 x 4 = 70 x 4 ∗tan − 8 − 2 x 5 ∗ sec 2 − 2 x 5 c) d dx
( tan
5
3 2 x^7 )= d
dx
( tan 2 x^7 )
5 3 =
∗tan 2 (^3 2) x^7 ∗ sec^2 2 x^7 ∗ 14 x^6 = 70 3 x 6 ∗tan 2 (^3 2) x^7 ∗ sec^2 2 x^7 d) d dx
( tan
− 3
4 2 x^9 )= d
dx
( tan 2 x^9 )
− 3 4 =
∗tan − 7 (^4 2) x^9 ∗ sec^2 2 x^9 ∗ 18 x^8 =−^54 4 x 8 ∗tan − 7 (^4 2) x^9 ∗ sec^2 2 x^9 =−^27 4 x 8 ∗tan − 7 (^4 2) x^9 ∗ s 4.- d dx
( cot
n
u ) =
d dx ( cot u ) n = n cot n − 1 u ∗− csc 2 u du dx =− n cot n − 1 u csc 2 u du dx
a) d dx
( cot
6 3 x 3
d dx
( cot 3 x
3
6 =− 6 cot 5 3 x 3 ∗ csc 2 3 x 3 ∗ 9 x 2 =− 54 x 2 ∗cot 5 3 x 3 ∗ csc 2 3 x 3 b) d dx
( cot
− 5 3 x 3
d dx
( cot 3 x
3
− 5 = 5 cot − 6 3 x 3 ∗ csc 2 3 x 3 ∗ 9 x 2 = 45 x 2 ∗cot − 6 3 x 3 ∗ csc 2 3 x 3 c) d dx
(cot
1 5 3 x
d dx
( cot 3 x^3 )
1 5 =
cot − 4 5 3 x 3 ∗ csc 2 3 x 3 ∗ 9 x 2 =−
x 2 ∗cot − 4 5 3 x 3 ∗ csc 2 3 x 3 d) d dx
(cot
− 1
2 3 x^3 )= d
dx
( cot 3 x^3 )
− 1 2 =
cot − 3 (^2 3) x^3 ∗ csc^2 3 x^3 ∗ 9 x^2 = 9 2 x 2 ∗cot 5 3 x 3 ∗ csc 2 3 x 3 5.- d dx
( sec
n
u ) =
d dx ( sec u ) n = n sec n − 1 u ∗ sec u ∗tan u du dx = n sec n u tan u du dx a) d dx
( sec
15 4 x 5
d dx
( sec 4 x
5
15 = 15 ∗ sec 15 4 x 5 ∗tan 4 x 5 ∗ 20 x 4 = 300 x 4 ∗ sec 15 4 x 5 ∗tan 4 x 5 b) d dx
( sec
− 9 2 x 7
d dx
( sec 2 x
7
− 9 =− 9 ∗ sec − 9 2 x 7 ∗tan 2 x 7 ∗ 14 x 6 =− 126 x 6 ∗ sec − 9 2 x 7 ∗tan 2 x 7 c) d dx
( sec
7 3 3 x
d dx
( sec 3 x^4 )
7 3 =
∗ sec 7 3 3 x 4 ∗tan 3 x 4 ∗ 12 x 3 =
x 3 ∗ sec 7 3 3 x 4 ∗tan 3 x 4 = 28 x 3 ∗ sec 7 3 3 x 4 ∗tan 3 x 4 d) d dx
( sec
− 5
7 2 x^6 )= d
dx
( sec 2 x^6 )
− 5 7 =
∗ sec − 5 (^7 2) x^6 ∗tan 2 x^6 ∗ 12 x^5 =−^60 7 x 5 ∗ sec − 5 (^7 2) x^6 ∗tan 2 x^6 ∗ 12 x^5 6.- d dx
( csc
n
u ) =
d dx ( csc u ) n = n csc n − 1 u ∗− csc u ∗cot u du dx =− n csc n u cot u du dx a) d dx
( csc
10 5 x 5
d dx
( csc 5 x
5
10 =− 10 ∗ csc 10 5 x 5 ∗cot 5 x 5 ∗ 25 x 4 =− 250 x 4 ∗ csc 10 5 x 5 ∗cot 5 x 5 b) d dx
( csc
− 4 5 x 5
d dx
( csc 5 x
5
− 4 = 4 ∗ csc − 4 5 x 5 ∗cot 5 x 5 ∗ 25 x 4 = 100 x 4 ∗ csc − 4 5 x 5 ∗cot 5 x 5 c) d dx
( csc
2
7 5 x^5 )= d
dx
( csc 5 x
5
2 7 =
∗ csc 2 (^7 5) x^5 ∗cot 5 x^5 ∗ 25 x^4 =−^50 7 x 4 ∗ csc 10 5 x 5 ∗cot 5 x 5 d) d dx
( csc
− 1
4 5 x^5 )= d
dx
( csc 5 x^5 )
− 1 4 =
∗ csc − 1 (^4 5) x^5 ∗cot 5 x^5 ∗ 25 x^4 = 25 4 x 4 ∗ csc 10 5 x 5 ∗cot 5 x 5 Derivada de funciones compuestas Modelo: d dx
( e
u
)= e
u du dx
