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UTFSM Matemáticas
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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
23. fx = log3x fx = 1x ln3
Aprendiendo a derivar
Sergio Yansen Nez
24. fx = log5x fx = 1x ln5
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
Aprendiendo a derivar
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
Aprendiendo a derivar
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
Aprendiendo a derivar
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
Aprendiendo a derivar
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
Aprendiendo a derivar
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
Aprendiendo a derivar
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