und. 19.- derivadas

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


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

= +



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


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



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= =



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


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


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



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

=



( )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


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 = + =



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= =



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


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

= =



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


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= =



( 6)f '' 0 > , existe mnimo en x = 6

(6)1 3f

6 36

= +

(6)3 1f

36 12

= =

CLAVE : D

und. 19.- derivadas

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