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• Mauricio España Fuentes

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Control de derivadas xxxxxxxxxx Fundamentos de Cálculo Instituto IACC 07 de Mayo de 2017

Desarrollo 1.- Desarrolle y determine el valor de la derivada usando definición 𝑓(𝑥) = 4𝑥 2 + 2𝑥 𝑑 (4𝑥 2 + 2𝑥) 𝑑𝑥 𝑑 𝑑 (4𝑥 2 ) + (2𝑥) 𝑑𝑥 𝑑𝑥 𝑑

𝑑

4𝑑𝑥 (𝑥 2 ) + 2 𝑑𝑥 (𝑥) 4*2𝑥 2−1 + 2 ∗ 1 8𝑥 + 2

2.- Calcule las siguientes derivadas inmediatas. 𝟒

𝒇(𝒙) = √𝟑𝒙𝟐 − 𝟓𝒙 + 𝟐 𝑑 𝑑 𝑑 [3𝑥 2 ] + [−5𝑥] + [2] 𝑑𝑥 𝑑𝑥 𝑑𝑥 3 ∗ 2𝑥 2−1 − 5 ∗ 1 + 0 6𝑥 − 5 + 0 6𝑥 − 5 4

𝑓(𝑥) = √3𝑥 2 − 5𝑥 + 2 𝑓´(𝑥) =

𝑓´(𝑥) =

(3𝑥 2 − 5𝑥 + 2) 4

4 ∗ √(3𝑥 2 − 5𝑥 + 2) 6𝑥 − 5 4

4 ∗ √(3𝑥 2 − 5𝑥 + 2)

𝑓(𝑥) =

(3𝑥 2 + 2𝑥) (3𝑥 − 1)

𝑑 𝑑 [3𝑥 2 ] + [2𝑥] 𝑑𝑥 𝑑𝑥 3 ∗ 2𝑥 2−1 + 2 ∗ 1 6𝑥 + 2 𝑑 𝑑 [3𝑥] + [−1] 𝑑𝑥 𝑑𝑥 3∗1+0 3

𝑓(𝑥) =

𝑓´(𝑥) =

(3𝑥 2 + 2𝑥) (3𝑥 − 1)

𝑢´ ∗ 𝑣 − 𝑢 ∗ 𝑣′ 𝑣2

(3𝑥 2 + 2𝑥)´ ∗ (3𝑥 − 1) − (3𝑥 2 + 2𝑥) ∗ (3𝑥 −)´ 𝑓´(𝑥) = (3𝑥 − 1)2 𝑓´(𝑥) =

(6𝑥 + 2) ∗ (3𝑥 − 1) − (3𝑥 2 + 2𝑥) ∗ (3) (3𝑥 − 1)2

(18𝑥 2 − 6𝑥 + 6𝑥 − 2) − (9𝑥 2 + 6𝑥) 𝑓´(𝑥) = (3𝑥 − 1)2 18𝑥 2 − 6𝑥 + 6𝑥 − 2 − 9𝑥 2 − 6𝑥 𝑓´(𝑥) = (3𝑥 − 1)2 9𝑥 2 − 6𝑥 − 2 𝑓´(𝑥) = (3𝑥 − 1)2

3.- Calcule f´(x) si f(x)= (3𝑥 4 − 18𝑥)5 ∗ 𝑐𝑜𝑠 (5𝑥 2 ) 𝑑 [(3𝑥 4 − 18𝑥)5 cos(5𝑥 2 )] 𝑑𝑥 (3𝑥 4 − 18𝑥)5 (3𝑥 4 − 18𝑥)5 (

𝑑 𝑑 [cos(5𝑥 2 )] + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 𝑑𝑥 𝑑𝑥

𝑑 𝑑 𝑑 [cos(𝑢1)] [5𝑥 2 ]) + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 𝑑𝑢 𝑑𝑥 𝑑𝑥

(3𝑥 4 − 18𝑥)5 (− sin(𝑢1)

𝑑 𝑑 [5𝑥 2 ]) + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

(3𝑥 4 − 18𝑥)5 (− sin(5𝑥 2 )

𝑑 𝑑 [5𝑥 2 ]) + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

(3𝑥 4 − 18𝑥)5 (− sin(5𝑥 2 ))

𝑑 𝑑 [5𝑥 2 ] + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

(3𝑥 4 − 18𝑥)5 ∗ − sin(5𝑥 2 )

𝑑 𝑑 [5𝑥 2 ] + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

−1 ∗ (3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) −1(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) −(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 )

𝑑 𝑑 [5𝑥 2 ] + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

𝑑 𝑑 [5𝑥 2 ] + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

𝑑 𝑑 [5𝑥 2 ] + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

−(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) (5

𝑑 2 𝑑 [𝑥 ]) + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

−5(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 )

𝑑 2 𝑑 [𝑥 ] + cos(5𝑥 2 ) [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥 𝑑𝑥

−5(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) (2𝑥) + cos(5𝑥 2 ) −10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + cos(5𝑥 2 ) −10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + cos(5𝑥 2 )(

𝑑 [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥

𝑑 [(3𝑥 4 − 18𝑥)5 ] 𝑑𝑥

𝑑 𝑑 [(𝑢2)5 ] [3𝑥 4 − 18𝑥]) 𝑑𝑢2 𝑑𝑥

−10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + cos(5𝑥 2 )(5𝑢24

𝑑 [3𝑥 4 − 18𝑥]) 𝑑𝑥

−10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + cos(5𝑥 2 )(5(3𝑥 4 − 18𝑥)4

𝑑 [3𝑥 4 − 18𝑥]) 𝑑𝑥

−10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + cos(5𝑥 2 )(5(3𝑥 4 − 18𝑥)4 −10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + 5cos(5𝑥 2 )(3𝑥 4 − 18𝑥)4 −10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + 5cos(5𝑥 2 )(3𝑥 4 − 18𝑥)4 (

𝑑 [3𝑥 4 − 18𝑥] 𝑑𝑥

𝑑 [3𝑥 4 − 18𝑥] 𝑑𝑥

𝑑 𝑑 [3𝑥 4 ] + [−18𝑥]) 𝑑𝑥 𝑑𝑥

−10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + 5cos(5𝑥 2 )(3𝑥 4 − 18𝑥)4 (3

𝑑 4 𝑑 [𝑥 ] + [−18𝑥]) 𝑑𝑥 𝑑𝑥

−10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + 5cos(5𝑥 2 )(3𝑥 4 − 18𝑥)4 (12𝑥 +

𝑑 [−18𝑥]) 𝑑𝑥

−10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + 5cos(5𝑥 2 )(3𝑥 4 − 18𝑥)4 (12𝑥 − 18

𝑑 [𝑥]) 𝑑𝑥

−10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + 5cos(5𝑥 2 )(3𝑥 4 − 18𝑥)4 (12𝑥 − 18 ∗ 1) −10(3𝑥 4 − 18𝑥)5 sin(5𝑥 2 ) 𝑥 + 5(3𝑥 4 − 18𝑥)4 cos(5𝑥 2 )(12𝑥 3 − 18)

Bibliografía IACC (2015). Derivadas. Fundamentos de cálculo. Semana 6