Derivadas De Orden Superior

ACTIVIDAD 5. DERIVADAS DE ORDEN SUPERIOR.

1. En un documento de Word, calcula las primeras tres derivadas de las siguientes funciones:

a) f(x) = 6x10 + x6 – 7x3

f ‘(x) = 60x9 + x5 – 21x2

f’’(x) = 540x8 + x4 – 42x

f ‘’’(x)= 4320 x7 + x3 – 42

b) f (x) = x3 – 2

x2 - 1

f ’ (x) = (x3 -2)’ (x2 + 1) – (x2 + 1)’ (x3 – 2)

(x2 + 1)2

f ‘(x) = (3x2) (x2 +1 ) – (2x) (x3 – 2)

(x2 +1)2

Simplificando:

f ‘(x) = 3x4 + 3x2 - 2x4 + 4x f ‘ (x) = x4 +3x2 + 4x

(x2 + 1)2 x4 +1

f ‘’(x) = (x4 + 3x2 + 4x)’ (x4 + 1) – (x4 +1)’ (x4 + 3x2 + 4x)

(x4 + 1)2

f ‘’ (x) = (4x3 + 6x +4) (x4 +1) – (4x3) (x4 +3x2 + 4x)

(x4 +1 )2

Simplificando:

f ’’(x) = 4x7 + 4x3 + 6x5 + 6x + 4x4 + 4 – 4x7 – 12x5 – 16x4

(x4 + 1)2

f ‘’(x) = 4x3 +6x – 12x5 – 12x4 +4

x16 + 1

f ‘’’(x) = (4x3 + 6x -12x5 -12x4 + 4)’ (x16 + 1) – (x16 + 1)’ (4x3 + 6x -12x5 -12x4 + 4)

(x16 + 1)2

f ‘’’(x) = (12x2 + 6 – 60x4 – 48x3) (x16 +1) – ( 16x5) (4x3 + 6x – 12x5 – 12x4 +4)

(x16 + 1)2

c) f (x) = 18 - 1 x6

2x6 4

f ‘ (x) = 9x6 - x6

f ‘ (x) = 54 x5 - x5

f ‘’ (x) = 270x4 - x4

f ‘’’(x) = 1080x3 - x3

d) F(x) = 5x6 – 2x3 –x

f ‘(x) = 30 x5 – 6x2 -1

f ‘’(x) = 150x4 – 12x

f ´´´(x) = 600x3 -12

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