Question

I. Deriva las siguientes funciones:
a) Utilizando primero las reglas básicas para derivar y despues simplificndo.
b) Simplificando primero (algebraicamente) y después aplicando las reglas correspondienes.

1) y = 4x$\sqrt{x}$
2) y = $\frac{9 x^{2} - 1 }{3x - 1}$

Answer 1

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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