Funciones

PRODUCTOS NOTABLES:
1. (a+b)²=a²+2ab+b²
2. (a-b)²=a²-2ab+b²
3. (a²-b²)²=a⁴-2a²b²+b⁴
4. (a+b)(a-b)=a²-b²
5. (a+b)³=a³+3a²b+3ab²+b³
6. (a-b)³=a³-3a²b+3ab²-b³
7. a³-b³=(a-b)(a²-ab+b²)
8. a³+b³=(a+b)(a²+ab+b²)

Lim ³√ x__
X → 4 -7x+1

³√ lim x = ³√ 4___X → 4 (-7) (4)+1
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-7 lim x + lim 1 = ³√ 4__ = ³√ 4_ = ³√ 4_ = -0.529
X → X → 4 -28+1 -27 -3

Lim ³√ 5+2x = ³√ 5+2(-3) = ³√ 5-6 = ³√ -1 = -1 = -0.5
X → -35-x 5-(-3) 5+3 8 2
*REGLA DE LA CADENA:

Si: f(x)= [g(x)]
f¹(x)= [g(x)]

f(x)= (5x²-2)⁹

(5x²-2)(5x²-2)(5x²-2)……. (5x²-2)

f¹(x) = 9(5x²-2)⁸10x

f(x)= 2x³+4
X²+1

f¹(x) =(x²+1)(6x²)-(2x³+4)(2x)
(X²+1)²

f¹(x)=6x⁴+6x²-[(2x³+4) (2x)]
(X²+1)²

f¹(x)=6x⁴+6x²-4x⁸-8x = 2x⁴+6x²-8x
X⁸+2x²+1 x⁴+2x²+1

h(x)= √ 5x²+3x+2h(x)= (5x²+3x+2)½

h¹(x)=½ (5x²+3x+2)¹¯½ (10x+3)

=½(5x²+3x+2)¯½ (10x+3)

= 10x+3____
2(5x²+3x+2)½

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f(x)= (x²+4)²(2x³-1)³

f(x) g(x)

= (x²+4)² Dx (2x³-1)³ + (2x³-1)³ Dx (x²+4)²

h¹(x)= (x²+4)² 3(2x³-1)² (6x²) + (2x³-1)³ + 2(x²+4)2x

(X²+4)²= x⁴+8x²+16
(2x³-1)²= (3x³)²+2(2x³)(-1)+(-1)²= 4x⁶-4x³+1
3(4x⁶-4x³+1)=12x⁶-12x³+3
(12x⁶-12x³+3)6x²= 72x⁸-72x⁵+18x²(X⁴+8x²+16)(72x⁸-72x⁵+18x²)= 72x¹²-72x⁹+18x⁶+576x¹⁰-576x⁷+144x²+1152x⁸-
1152x⁵+288x²

*Ejercicios:1. lim t + 2
T ->2 t²-4

R= t+2__ = 1_ = IND
(t+2)(t-2) t-2

2. lim x+2
x -> 1 1-x

R= IND

3. lim x²+x-6_
x -> 2 x²-6x+8

R= 4+2-6_ = 0 x²+x-6= (x+3) (x-2)
4-12+8 0 x²-6x+8= (x-4) (x-2)
(X-4)(x-2) lim (x+3) (x-2) = X+3 = 5_ = 2.5X²-2x-4x+8 x -> 2 (x-4) (x-2) x-4 -2
X²-6x+8

4. Dx [ 2x+1 (3x-1) ]
X+5

R= a ( 1 )= a
b b

(2x+1) (3x-1)= 6x²-2x+3x-1 = (x+5) Dx (6x²+x-1)-(6x²+x-1) Dx (x+5) =
X+5 x+5 (x+5)²

(x+5)(12x+1)-(6x²+x-1) (1) = 12x²+x+60x+5-[6x²+x-1] = 12x²+61x+5-6x²-x+1 =
(X+5)² x²+10x+25 x²+10x+25

6x²+60x+6 = x²+10x+1 _ = (x+0.1010) (x+9.8989)
X²+10x+25 x²+10x+25 (x+5) (x+5)

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