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UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 1 -
Captulo 19.1. Derivadas de una funcin
01. (x)
(x h) (x)(x h)
5x 2f 5h3 f f5x 5h 2 3f
3
+
+
+ =
=+ +
=
(x h) (x)f f 5h 3
+ =
(x h) (x)h 0 h 0
f f 5Lm Lmh 3
+
=
(x)5f '3
=
CLAVE : A
02. Correccin de clave:
2
13A)-8x +8x+2
(x)3x 5f4x 2
+=
(x h)3x 3h 5f4x 4h 2+
+ +=
+
(x h) (x) 226hf f
16x (16h 16)x 8h 4+ =
+ +
(x h) (x)2
f f 13h 8x (8h 8)x 4h 2
+ =
+ +
Cuando h = 0
(x) 213f '
8x 8x 2=
+
CLAVE : A
03. (x)f 4x 1=
(x h)f 4x 4h 1+ = +
(x h) (x)f f 4x 4h 1 4x 1+ = +
(x h) (x)4x 4h 1 4x 1f f4x 4h 1 4x 1+
+ + =
+ +
(x h) (x)4hf f
4x 4h 1 4x 1+ =
+ +
(x h) (x)f f 4h 4x 4h 1 4x 1
+ =
+ +
Cuando h = 0:
(x)4 4f '
4x 1 4x 1 2 4x 1= =
+
(x)2f '
4x 1=
CLAVE : C
04. 3(x)f x 5=
3(x h)f x h 5+ = +
3 3(x h) (x)f f x h 5 x 5+ = +
( ) ( )(x h) (x) 2 23 3 3 3x h 5 x 5f f
x h 5 x h 5 x 5 x 5+
+ + =
+ + + +
( ) ( )(x h) (x) 2 23 3 3 3hf f
x h 5 x h 5 x 5 x 5+ =
+ + + +
( ) ( )(x h) (x)
2 23 3 3 3
f f 1h
x h 5 x h 5 x 5 x 5
+ =
+ + + +
Cuando h = 0
( ) ( ) ( )(x) 2 2 23 3 31f '
x 5 x 5 x 5=
+ +
( )(x) 231f '
3 x 5=
CLAVE : A
05. (x)f ' 10x 3= (3)f ' 27=
L : y = 27x + b 37 = 81 + b b= 44
L: y=27x 44
CLAVE : C
06. Correccin de clave:
B) 8x-5y+1=0
12 2(x)f (3x 2x 4)= +
12 2(x)
1f ' (3x 2x 4) (6x 2)2
= +
(x) 23x 1f '
3x 2x 4
=
+
(3)8 8f '
525= =
L: 8y x b5
= +
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 2 -
24 15 b b5 5
= + =
L: 8 1y x5 5
= +
L: 8x 5y + 1 = 0 CLAVE : B
07. 75 23(x)f 12 x 32 x 8x=
75232(x)f 12x 32x 8x=
4332(x)
5 7f ' 12 x 32 x 16x2 3
=
43 3(x)224f ' 30 x x 16x3
=
3(x)224f ' 30x x x x 16x
3=
CLAVE : A
08. 5 1113 6(T)f 5 T 10T 35 T
= +
5 1113 6(T)f 5T 10T 35T
= +
8 1723 6(T)
25 385f ' T 10T T3 6
= +
8 172 3 6(T)
60T 50 T 385 Tf '6
=
CLAVE : A
09. 7 533 3 6(x)f 21x 3 x 7 x x= +
7 533 3 62(x)f 21x 3x 7x x= +
4 112 3 62(x)
21 5f ' 63x 7x x x2 6
= +
12 3 6(x)21 5f ' 63x 7x x x x2 6
= +
CLAVE : B
10. 14 95 35 5(z)f 20 z 4 z 15 z 3 z= +
14 95 35 52 2(z)f 20z 4z 15z 3z= +
9 43 15 52 2(z)
56 45 27f ' 50z z z z5 2 5
= +
4 45 5(z)56 45 27f ' 50z z z z z z5 2 5
= +
CLAVE : A
11. Correccin de clave:
B) -5
(x)f 8 5x=
(x)f ' 5 f '( 1) 5= =
CLAVE : B
12. 2(x)f 3x 4= +
(x) (2)f ' 6x f ' 12= =
CLAVE : C
13. 2(x)f 3x 2x 1= +
(x) (1)f ' 6x 2 f 4= =
CLAVE : D
14. 2(x)f 6 3x x=
(x) (0)f ' 3 2x f ' 3= =
CLAVE : A
15. 3(x)f 8 x=
2(x) ( 2)f ' 3x f ' 12= =
CLAVE : A
16. 12(x)f x=
12(x)
1 1f ' x2 2 x
= =
(4)1 1f '
2(2) 4= =
CLAVE : C
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 3 -
17. 12(x)f 3x 5 (3x 5)= + = +
12(x)
1f ' (3x 5) (3)2
= +
(x)3f '
2 3x 5=
+
(0) (0)3 3 5f ' f '
102 5= =
CLAVE : A
18. 1(x)f (x 1)= + 2(x)f ' (x 1) (1)= +
(x) 21f '
(x 1)=
+ ( 2)
1f ' 1(1) = =
CLAVE : C
19. 2(x)f x x
=
3(x)f ' 2x 1
=
(1)f ' 2 1 3= =
CLAVE : C
20. 2 1(x)f 3(x 1)= + 2 2(x)f ' 3(x 1) (2x)= +
(x) 2 26xf '
(x 1)=
+ (0)f ' 0=
CLAVE : C
21. 2(x)f 1 x= (x) (3)f ' 2x f ' 6= =
CLAVE : D
22. 1(x)4f x5
= 2(x) (2)4 1f ' x f '5 5
= =
CLAVE : B
23. 12(x)f 2x 1
=
32(x)f ' x
=
(x) (4)1 1f ' f '
8x x= =
CLAVE : D
24. 12(x)f (9x 1)= +
(x) (7)9 9 9f ' f '
2(8) 162 9x 1= = =+
CLAVE : B
25. 12(x)f (2x 3)
= +
32(x)
1f ' (2x 3) (2)2
= +
(x) (3)1 1f ' f '
9(3)(2x 3) 2x 3= = + +
(3)1f '
27=
CLAVE : E
26. 13(x)f x x
=
43(x)
1f ' x 13
=
(x) ( 8)31 1f ' 1 f ' 1( 8)( 2)x x
= =
( 8)1f ' 1
16= ( 8)
17f '16
=
CLAVE : A
27. 2y 4x 1= +
y ' 8x m 8(1)= = m = 8
L: y = mx + b
Y = 8x+b
5 = 8 + b b = 3
L: 8x y 3 = 0
CLAVE : A
28. 32y 4x=
123y ' 4 x
2=
y ' 6 x m 6 1= =
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 4 -
m = 6
L: y = 6x + b
4 = 6 + b b = 2
L: 6x y 2 = 0
CLAVE : B
29. 4(x)f 3x 2x= + 3
(x)f ' 12x 2= +
CLAVE : D
30. Correccin de clave:
2
2
x -2x+1B)x 2 1x +
2 2(x) (x) 2
x 1 (2x)(x 1) (1)(x 1)f f 'x 1 (x 1)
+ += =
2(x) 2
x 2x 1f 'x 2x 1
=
+
CLAVE : B
31. Correccin de clave:
C) 18x+8
3 2(x)h 3x 4x x= +
2(x)h' 9x 8x 1= +
[ ]x
d h'(x) 18x 8d
= +
CLAVE : C
32. 2(x)f x 2x= +
(x) (1)f ' 2x 2 f ' 4= + =
(x) (2)f '' 2 f '' 2= =
(1) (2)f ' f '' 6+ =
CLAVE : D
33. 3(x)f (4x)=
3 2(x) (x)f 64x f ' 192x= =
CLAVE : B
34. { }(x) (f )f 1 x; Dom 0= + = { }(x) (f )f ' 1; Dom 0= = f(0) no existe
CLAVE : A
35. 12(x)f 2x 3x=
12(x)
3f ' 2 x2
=
(x) (1)
(x)
3 3f ' 2 f ' 222 x
1f '2
= =
=
CLAVE : D
Captulo 19.2. Regla de cadena 01. Segn la teora I) V II) V III) V
CLAVE : A
02. Segn la teora I) V II) V III) V
CLAVE : C
03. Correccin de enunciado:
Calcular 8
6
a
b
+, si la funcin:
Por condicin (x)f es continua.
3 21 1 1a 4 b 3
2 2 2
+ =
a b1 38 2
+ = +
a b1 38 2
= +
a 8 = 4b + 24 a 8 = 4(b + 6)
a 8 4b 6
=
+
CLAVE : D
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 5 -
04. (x)f (x 1)(x 2)(x 3)(x 4)= + + + +
4 3 2(x)f x 10x 35x 50x 24= + + + +
3 2(x)f ' 4x 30x 70x 50= + + +
( 3)f 108 270 210 50 = + +
( 3)f ' 2 =
CLAVE : C
05. (x)f (x a)(x b)(x c)= 3 2
(x)f x (a b c)x (ab ac bc)x abc= + + + + + 2
(x)f ' 3x 2(a b c)x (ab ac bc)= + + + + + 2
(a)f ' 3a 2(a b c)(a) ab ac bc= + + + + + 2
(a)f ' a ab ac bc a(a b) c(a b)= + =
(a)f ' (a b)(a c)=
CLAVE : A
06. (x)1 af x
a b a b=
(x)1f '
a b=
(a)1f '
a b=
CLAVE : B
07. b a(x)f (1 ax )(1 bx )= + +
a b a b(x)f 1 bx ax abx
+= + + +
a 1 b 1 a b 1(x)f ' abx abx ab(a b)x + = + + +
(1)f ' ab ab ab(a b)= + + + (1)f ' ab(2 a b)= + +
CLAVE : C
08. 9 3(x)f (x 2)= +
9 8(x)f ' 3(x 2)(9x )= +
( 1) ( 1)f ' 3(1)(9) f ' 27 = =
CLAVE : C
09. 11 132 4(x)f x x x= + +
21 332 4(x)
1 1 1f ' x x x2 3 4
= + +
(1)1 1 1f '2 3 4
= + +
(1)6 4 3 13f '
12 12+ +
= =
CLAVE : C
10. 12(x)f (x 2)= +
12(x)
1f ' (x 2) (1)2
= +
(x)1f '
2 x 2=
+
(3) (3)1 5f ' f '
102 5= =
CLAVE : E
11. 2
(x) 2x 3xfx 1
+=
2 2(x) 2 2
(2x 3)(x 1) (2x)(x 3x)f '(x 1)
+ +=
3 2 3 2(x) 2 2
2x 2x 3x 3 2x 6xf '(x 1)
+ =
2(x) 2 2
3x 2x 3f '(x 1)
=
CLAVE : E
12.
13 3
(x) 2x xf
x 2x 1
+=
+
23 2 2 33
(x) 2 2 21 x x (3x 1)(x 2x 1) (2x 2)(x x)f ' 3 x 2x 1 (x 2x 1)
+ + + +=
+ +
23
(2)1 10 (13)(1) (2)(10)f '3 1 1
=
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 6 -
( )2
2 33(2) (2)
1 7 1f ' 10 ( 7) f '3 3 10
= =
2 23 3(2)
7 100 7 1f ' (100)3 1000 3 100
= =
3(2)7f ' 10
30=
CLAVE : E
13. (x)ax 1fx 2
=
(x) 2a(x 2) (1)(ax 1)f '
(x 2)
=
(x) 2a(x 2) (ax 1)f '
(x 2)
=
Condicin: 3a 5a 1 19
+=
2a 1 9 + =
a = 5
CLAVE : D
14. Por condicin
2 3 4(x) 0 1 2 3 4P a a x a x a x a x + + + +
2 3(x) 1 2 3 4P' a 2a x 3a x 4a x + + +
De donde:
1 2 3 41
a 0 a 0 a 1 a2
= = = =
Ahora: 3 4(x) 01P a x x2
+ + (1) 03P a2
= +
CLAVE : E
15. (x)f x x= +
11 22(x)f x x
= +
11 122 2(x)
1 1f ' x x 1 x2 2
= + +
12(1)
1 1 1 3f ' (2) 12 2 22 2
= + =
(1)3 2f '
8=
CLAVE : B
16. 2 2
(x) 2 2a xfa x
+=
2 2 2 2(x) 2 2 2
(2x)(a x ) ( 2x)(a x )f '(a x )
+=
2 3 2 3(x) 2 2 2
2a x 2x 2a x 2xf '(a x )
+ +=
2 3(x) (2a)2 2 2 4
4a x 8af ' f '(a x ) 9a
= =
(2a)8f '9a
=
CLAVE : A
17. (x)xf
x a=
+
(x) 2 2(x a) (1)(x) af '
(x a) (x a)+
= =
+ +
(a) 2 2a 1f '
4a a= =
a = 4
CLAVE : D
18. Correccin de clave:
E) N.A.
2(x) 2
x 2ax bfx a
+=
+
2 2(x) 2 2
(2x 2a)(x a) (2x)(x 2ax b)f '(x a)
+ +=
+
2 2(x) 2 2
2ax 2ax 2bx 2af '(x a)
+ =
+
Condicin: 2
(2) 28a 4a 4b 2af ' 0 0
(a 4)+
= =+
2 212a 4b 2a 0 a 6a 2b = + =
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 7 -
Condicin: 2
(1) 22a 2a 2b 2af ' 2 2
(a 1)+
= =+
2 24a 2b 2a 2a 4a 2 = + +
2 22b 4a 2 2 4a 2b = = ..
De y : 2 2a 6a 2 4a + =
23a 6a 2 0+ + =
26 (6) 4(6) 6 2 3a
6 6
= =
3 3 3 3a a
3 3 +
= =
2 24 2 3 4 2 3a a3 3
+= =
En : 2b (1 2a )= +
11 4 3 11 4 3b b3 3
+= =
15 11 3 15 11 3
ab ab3 3
+= =
NO HAY CLAVE
19. 2
(x)(x) (x)7
(x)
f 3x 2xH g(f )
g x
= +
==
2 7(x)H (3x 2x)= +
2 6(x)H' 7(3x 2x) (6x 2)= + +
6( 1)H' 7(1) ( 4) 28 = =
CLAVE : E
20. (x)x 1g fx 1
+ =
Sea: xc (x)x 1hx 1
+=
Ahora: (x)g f(h(x))= (x)g' f '(h(x)) h'(x)= ..(1)
Tambin: 22h'(x)
(x 1)
=
En (1): x = 0 (0)g' f '(h(o))h'(o)=
(0)g' f '( 1) ( 2 )= (0)g' 2( 2 ))= (0)g' 4=
CLAVE : D
Captulo 19.3. Extremos relativos
01. 2(t)S 16t 96t 12= + +
(t)S' 32t 96= +
(t)S' 0 t 3= =
(t)S 16(9) 96(3) 12 156= + + =
CLAVE : C
02. Correccin de enunciado:
2( ) 90 mn 2025f x x x f= + =
En cada caso:
I) 2(x)f x 90x= + (x)f ' 2x 90= +
(x)f ' 0 x 45= =
( 45)f ' min 2025 = =
II) 2(x)g x 60x= + (x)g' 2x 60= +
(x)g' 0 x 30= =
(30)g mx 900= =
III) 2(x)h 28x x= (x)h' 28 2x=
(x)h' 0 x 14= =
(14)h' mx 196= =
VVF
CLAVE : D
03. 2(x)f 18x x=
(x)f ' 18 2x=
(x)f ' 0 x 9= =
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 8 -
2 2 2(9)f 2(9) 9 9 81= = =
Observa que: a 9 b 81= = a + b = 90
CLAVE : B
04. 2(x)f x 12x= +
(x)f ' 2x 12= +
(x)f ' 0 x 6= =
( 6)f 36 72 36 = =
Observa que: m 6 n 36= =
2 2m n 1332+ =
CLAVE : E
05. 2(x)f x 18x 89= +
2y (x 9) 8= + Vrtice= (9; 8) a 9 b 8= =
E ab ba aob= + + E = 98 + 89 + 908
E = 1095
CLAVE : A
06. Correccin de clave:
B) mn(2;-1)
2(x)f x 4x 3= +
(x)f ' 2x 4= (x)f ' 0 2x 4 0= =
x = 2
(2)f min 1= =
Rpta: (2; 1) CLAVE : B
07. 3(x)f x 12x 12= +
2(x)f ' 3x 12=
(x)f ' 0 x 2 x 2= = =
(x)f '' 6x=
(2)f '' 12 0= > existe mnimo en x = 2
( 2)f '' 12 0 = < existe mximo en x = 2
i) Si x = 2 (2)f 4= ii) Si x = 2 ( 2)f 28 =
Rpta: mn(2; 4) y mx(2; 28) CLAVE : C
08. 3 2(x)f x 3x 2= +
2(x)f ' 3x 6x=
(x)f ' 0 x 0 x 2= = =
(x)f '' 6x 6=
(0)f '' 6 0= < existe mximo en x = 0
(2)f '' 6 0= > existe mnimo en x = 2
i) Si x = 0 (0)f 2= ii) Si x = 2 (2)f 2=
Rpta: mx (0; 2) y mn (2; -2)
CLAVE : A
09. 3(x)f 40 30x 10x= +
2(x)f ' 30 30x=
(x)f ' 0 x 1 x 1= = =
(x)f '' 60x=
(1)f '' 60 0= < existe mximo en x = 1
( 1)f '' 60 0 = > existe mnimo en x = 1
i) Si x = 1 (1)f 60= ii) Si x = 1 ( 1)f 20 =
Rpta: mx (1; 60) y mn (1; 20) CLAVE : C
10. 1(x)f x 100x
= +
2(x) (x) 2
100f ' 1 100x f ' 1x
= =
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 9 -
3(x) (x) 3
200f '' 200x f ''x
= =
(x)f ' 0 x 10 x 10= = =
(10)f '' 0> existe mnimo en x = 10
( 10)f '' 0 < existe mximo en x = 10
i) Si x = 10 (10)f 20= ii) Si x = 10 ( 10)f 20 =
Rpta: mx(10; 20) y min(10; 20) CLAVE : B
11. 3(x)f 4 3x x= +
2(x)f ' 3 3x=
(x)f '' 6x=
(x)f ' 0 x 1 x 1= = =
En x = 1 existe mximo (1)f 6= : c = 1 d = 6
En x = 1 existe mnimo ( 1)f 2 = : a = 1 b = 2
2 2 2 2a b c d 42+ + + =
CLAVE : B
12. 1(x)f x 16x
= +
2(x) (x) 2
16f ' 1 16x f ' 1x
= =
3(x) (x) 3
32f '' 32x f ''x
= =
(x)f ' 0 x 4 x 4= = =
En x = 4 existe mnimo f(4) = 8: c = 4 d = 8
En x = 4 existe mximo f(4) = 8: a = 4 b = 8 Final:
E = 2a(b + c + d) E = 8(8 + 4 + 8) E = 32 CLAVE : B
13. 2 2
(x) 2 1/22x xf
(x 8)x 8= =
2 2 3 3(x) 2 2 2 2
2x x 8 x 8 x x 16xf '(x 8) x 8 (x 8) x 8
= =
(x)f ' 0= x = 0; x = 4; x = 4
(x)min16 16f 4 2
16 8 18= = =
CLAVE : D
14. 3 2(x)2f x 4x 6x 23
= + +
2(x)f ' 2x 8x 6= +
(x)f '' 4x 8=
(x)f ' 0 x 1 x 3= = =
En x = 1 existe mximo (1)14f3
=
En x = 3 existe mnimo (3)f 2=
CLAVE : B
15. 2(x)f x 2x 1= +
(x)f ' 2x 2=
(x)f ' 0 x 1= =
(x)mnf 0;= Luego (f )Ran 0',=
CLAVE : B
16. Correccin de clave:
1D) -
12
1 2(x)f x 3x
= +
2 3(x)f ' x 6x
=
3 4(x)f '' 2x 18x
= +
(x)f ' 0 x 6= =
UNIDAD 19: DERIVADAS lgebra Nivel Pre
Prof. Juan Carlos Ramos Leyva - 10 -
( 6)f '' 0 > , existe mnimo en x = 6
(6)1 3f
6 36
= +
(6)3 1f
36 12
= =
CLAVE : D