I. Deriva las siguientes funciones :a) Utilizando primero las reglas básicas para derivar y despues simplificndo?

1) y = 4x * √x

Simplificando y derivando :

y = 4x * √x

y = (4)(x) * [ (x) ^ 1 / 2 ]

y = (4) * (x) ^ (1 / 2 + 1)

y = 4 * (x) ^ (3 / 2)

dy / dx = (3 / 2) * (4) * (x) * ^ (3 / 2 - 1)

dy / dx = (2 * 3)(x) ^ (3 - 2) / 2

dy / dx = 6x ^ (1 / 2)

dy / dx = 6√x

Derivando y simplificando :

dy / dx = 4√x + (4x) * (1 / 2) * (x) ^ [ (1 / 2 - 1) ]

dy / dx = 4√x + 2x * (x) ^ ( - 1 / 2)

dy / dx = 4√x + 2 * (x) ^ (1 - 1 / 2)

dy / dx = 4√x + 2 * (x) ^ (1 / 2)

dy / dx = 4√x + 2√x

dy / dx = 6√x

2) y = ( 9x ^ 2 - 1) / (3x - 1)

Simplificando y derivando :

y = [ (3x - 1) * (3x + 1) / (3x - 1) ]

y = 3x + 1

dy / dx = 3

Derivando y simplificando :

dy / dx = [ (2 * 9x)(3x - 1) - (3) * (9x ^ 2 - 1) ] / (3x - 1) ^ 2

dy / dx = [ 18x (3x - 1) - (27x ^ 2 - 3) ] / ( 3x - 1) ^ 2

dy / dx = ( 54x ^ 2 - 18x - 27x ^ 2 + 3 ) / ( 3x - 1) ^ 2

dy / dx = ( 27x ^ 2 - 18x + 3) / (3x - 1) ^ 2

dy / dx = 27 ( x ^ 2 - 2 / 3 x + 1 / 9 ) / (3x - 1) ^ 2

dy / dx = 27 [ x ^ 2 - 2 / 3x + 1 / 9 + (2 / 3 * 2) ^ 2 - (2 / 3 * 2) ^ 2] / (3x - 1) ^ 2

dy / dx = 27 [ x ^ 2 - 2 / 3 x + 1 / 9 + (1 / 3) ^ 2 - (1 / 3) ^ 2 ] / (3x - 1) ^ 2

dy / dx = 27 (x ^ 2 - 2 / 3 x + 1 / 9 + 1 / 9 - 1 / 9 ) / ( 3x - 1) ^ 2

dy / dx = 27 ( x - 1 / 3) ^ 2 / ( 9x ^ 2 - 6x + 1 )

dy / dx = 27 ( x - 1 / 3) ^ 2 / [ 9 * (x ^ 2 - 6 / 9 x + 1 / 9 ) ]

dy / dx = 27 ( x - 1 / 3 ) ^ 2 / [ 9 * (x ^ 2 - 2 / 3 x + 1 / 9) + (2 / 3 * 2) ^ 2 - (2 / 3 * 2) ^ 2 ]

dy / dx = 27 ( x - 1 / 3) ^ 2 / [ 9 * (x ^ 2 - 2 / 3 x + 1 / 9 ) + (1 / 3) ^ 2 - (1 / 3) ^ 2]

dy / dx = 27 ( x - 1 / 3) ^ 2 / [ 9 * ( x ^ 2 - 2 / 3 x + 1 / 9) + (1 / 9) - (1 / 9)]

dy / dx = 27 ( x - 1 / 3) ^ 2 / [ 9 * (x - 1 / 3) ^ 2]

dy / dx = 27 / 9

dy / dx = 3

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