aprendiendo a derivar

Sergio Yansen Nez

Derivando funciones elementales:

1. fx = 4 fx = 0

2. fx = 3 fx = 0

3. fx = 14 fx = 0

4. fx = 5 fx = 0

5. fx = x fx = 1

6. fx = x2 fx = 2x

7. fx = x3 fx = 3x2

8. fx = x4 fx = 4x3

9. fx = x5 fx = 5x4

10. fx = x6 fx = 6x5

11. fx = x1 fx = 1x11 = x2

12. fx = x2 fx = 2x21 = 2x3

13. fx = x3 fx = 3x31 = 3x4

14. fx = x 12 fx = 12 x12 1 = 12 x

12

15. fx = x 25 fx = 25 x25 1 = 25 x

35

16. fx = x 54 fx = 54 x

54 1 = 54 x

94

17. fx = ex fx = ex

18. fx = 2x fx = 2x ln2

19. fx = 3x fx = 3x ln3

20. fx = 5x fx = 5x ln5

21. fx = lnx fx = 1x

22. fx = log2x fx = 1x ln2


Aprendiendo a derivar

Sergio Yansen Nez


25. fx = x fx = 12 x

26. fx = 3 x fx = 13 3 x2

27. fx = 4 x fx = 14 4 x3

28. fx = 5 x fx = 15 5 x4

29. fx = 6 x fx = 15 5 x4

30. fx = 7 x fx = 17 7 x6


Sergio Yansen Nez

Propiedad: fx gx = fx gx

1. fx = 2 + x x2 + x4

fx = 0 + 1 2x + 4x3 = 1 2x + 4x3

2. fx = 5 + lnx + 3x

fx = 0 + 1x + 3x ln3 = 1x + 3x ln3

3. fx = x3 + 3x 1

fx = 3x2 + 3x ln3 0 = 3x2 + 3x ln3

4. fx = ex x + 1

fx = ex 1 + 0 = ex 1

5. fx = x x4 + x

fx = 1 4x3 + 12 x

6. fx = x7 x7 + 7x 7

fx = 7x6 + 7x8 + 7x ln7 0 = 7x6 + 7x8 + 7x ln7

7. fx = ex + x2 x + x

fx = ex + 2x 12 x

+ 1


Sergio Yansen Nez

Propiedad: cons tan te fx = cons tan te fx

1. fx = 2x3 3x4 + 3 2x + 4

fx = 2 3x2 3 4x3 + 3 2x ln2 + 0 = 6x2 12x3 + 3 2x ln2

2. fx = 3x2 5 lnx 7x 1

fx = 3 2x 5 1x 7 1 0 = 6x 5x 7

3. fx = 7x3 + 3ex 4

fx = 7 3x4 + 3 ex 0 = 21x4 + 3ex

4. fx = 4x 2 lnx + 6

fx = 4 1 2 1x + 0 = 4 2x = 4 2x

5. fx = 5 5x 5 x

fx = 0 5 1 5 12 x

= 5 52 x


Sergio Yansen Nez

Propiedad: fx gx = fx gx + fx gx

1. fx = x2 lnx

fx = x2 lnx + x2 lnx

fx = 2x lnx + x2 1x = 2x lnx + x

2. fx = x3ex

fx = x3 ex + x3 ex

fx = 3x2ex + x3ex

3. fx = ex lnx

fx = ex lnx + ex lnx

fx = ex lnx + ex 1x = ex lnx + e

xx

4. fx = lnx x

fx = lnx x + lnx x

fx = 1x x + lnx 12 x

=xx +

lnx2 x

=1x

+lnx2 x

5. fx = ex x

fx = ex x + ex x

fx = ex x + ex 12 x

= ex x + ex

2 x


Sergio Yansen Nez

Propiedad fxgx

=fx gx fx gx

gx2

1. fx = x2

lnx

fx = x2

lnx x2 lnx

lnx2

fx = 2x lnx x2 1x

lnx2=2x lnx xln2x

2. fx = x3

ex

fx = x3

ex x3 ex

ex2

fx = 3x2ex x3exe2x

=ex3x2 x3

e2x=3x2 x3ex

3. fx = ex

lnx

fx = ex lnx ex lnx

lnx2

fx =ex lnx ex 1x

lnx2=ex lnx e

xx

ln2x=xex lnx exx ln2x

=exx lnx 1x ln2x

4. fx = lnxx

fx = lnx x lnx x

x 2

fx =

1x x lnx

12 x

x 2=

xx

lnx2 xx =

2 lnx2x x

5. fx = ex

x

fx = ex x ex x

x 2


Sergio Yansen Nez

fx =ex x ex 1

2 x x 2

=

ex x ex

2 xx =

2exx ex2x x

=ex2x 12x x

Regla de la cadena: fgx = fgx gx

1. fx = x3 + 4x 35

fx = 5x3 + 4x 34 x3 + 4x 3

fx = 5x3 + 4x 343x2 + 4

2. fx = 2x5 5x 27

fx = 72x5 5x 26 2x5 5x 2

fx = 72x5 5x 2610x4 5

3. fx = x4 3x + 6

fx = 12 x4 3x + 6

x4 3x + 6

fx = 12 x4 3x + 6

4x3 3 = 4x3 3

2 x4 3x + 6

4. fx = x5 + x + 1

fx = 12 x5 + x + 1

x5 + x + 1

fx = 12 x5 + x + 1

5x4 + 1 = 5x4+ 1

2 x5 + x + 1

5. fx = ex3+5x2

fx = ex3+5x2 x3 + 5x 2

fx = ex3+5x2 3x2 + 5 = ex3+5x23x2 + 5

6. fx = e2x53x2+x2

fx = e2x53x2+x2 2x5 3x2 + x 2

fx = e2x53x2+x2 10x4 6x + 1 = e2x53x2+x210x4 6x + 1

7. fx = lnx5 2x + 7


Sergio Yansen Nez

fx = 1x5 2x + 7

x5 2x + 7

fx = 1x5 2x + 7

5x4 2 = 5x4 2

x5 2x + 7

8. fx = ln2x4 x + 5

fx = 12x4 x + 5

2x4 x + 5

fx = 12x4 x + 5

8x3 1 = 8x3 1

2x4 x + 5


aprendiendo a derivar

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