143:@0.937671:0.974518:0.968375:0.974518:0.968375:0.956333:0.937671:0.956333:0.000000:0.000000:0.000000
Pensamiento numérico y variacional:@0.603971:0.043660:0.926928:0.043660:0.926928:0.024915:0.603971:0.024915:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
6.1 Derivada de la función exponencial :@0.073089:0.136836:0.381513:0.136836:0.381513:0.119889:0.073089:0.119889:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Sea  ( ) =  . Encontremos la derivada de la función utilizando la definición de derivada::@0.073089:0.164198:0.702811:0.164199:0.702811:0.147766:0.073089:0.147764:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.102706:0.164198:0.119586:0.164198:0.119586:0.147792:0.102706:0.147792:0.000000:0.000000:0.000000
a:@0.143092:0.164198:0.152350:0.164198:0.152350:0.147792:0.143092:0.147792:0.000000
x:@0.152307:0.158135:0.156654:0.158135:0.156654:0.148571:0.152307:0.148571:0.000000
 = :@0.650066:0.197503:0.668234:0.197503:0.668234:0.181069:0.650066:0.181069:0.000000:0.000000:0.000000
a .lna:@0.667542:0.197503:0.706347:0.197503:0.706347:0.181097:0.667542:0.181097:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.676650:0.191439:0.680996:0.191439:0.680996:0.181875:0.676650:0.181875:0.000000
Teniendo en cuenta que :@0.073083:0.232174:0.254460:0.232174:0.254460:0.215740:0.073083:0.215740:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
  (límite que se puede evaluar al tabular :@0.419462:0.232179:0.708183:0.232179:0.708183:0.215745:0.419462:0.215745:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
valores cercanos a cero por derecha e izquierda). :@0.073089:0.256133:0.428288:0.256133:0.428288:0.239699:0.073089:0.239699:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Conclusión::@0.073089:0.283495:0.161606:0.283495:0.161606:0.266825:0.073089:0.266825:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 si  ( ) =  , entonces  '( ) =  .:@0.161480:0.283495:0.376295:0.283496:0.376295:0.267062:0.161480:0.267061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.179925:0.283495:0.196804:0.283495:0.196804:0.267089:0.179925:0.267089:0.000000:0.000000:0.000000
a:@0.220310:0.283495:0.229569:0.283495:0.229569:0.267089:0.220310:0.267089:0.000000
x:@0.229533:0.277432:0.234136:0.277432:0.234136:0.267851:0.229533:0.267851:0.000000
f  x:@0.312504:0.283496:0.335826:0.283496:0.335826:0.267090:0.312504:0.267090:0.000000:0.000000:0.000000:0.000000
a lna:@0.359332:0.283496:0.398623:0.283496:0.398623:0.267090:0.359332:0.267090:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.368536:0.277432:0.373140:0.277432:0.373140:0.267851:0.368536:0.267851:0.000000
.:@0.398568:0.283496:0.401771:0.283496:0.401771:0.267062:0.398568:0.267062:0.000000
De manera que si tenemos  ( ) =  , entonces   '( ) =  .:@0.073091:0.310858:0.466725:0.310858:0.466725:0.294424:0.073091:0.294424:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.272607:0.310858:0.289487:0.310858:0.289487:0.294452:0.272607:0.294452:0.000000:0.000000:0.000000
e:@0.312993:0.310858:0.320779:0.310858:0.320779:0.294452:0.312993:0.294452:0.000000
x:@0.320732:0.304794:0.325336:0.304794:0.325336:0.295213:0.320732:0.295213:0.000000
f x:@0.403704:0.310858:0.427725:0.310858:0.427725:0.294452:0.403704:0.294452:0.000000:0.000000:0.000000
e lne  e:@0.451231:0.310858:0.513406:0.310858:0.513406:0.294452:0.451231:0.294452:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.458966:0.304794:0.463569:0.304794:0.463569:0.295213:0.458966:0.295213:0.000000
=  .1 = :@0.490783:0.310858:0.548810:0.310858:0.548810:0.294424:0.490783:0.294424:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.513329:0.304794:0.517933:0.304794:0.517933:0.295213:0.513329:0.295213:0.000000
e:@0.548699:0.310858:0.556486:0.310858:0.556486:0.294452:0.548699:0.294452:0.000000
x:@0.556420:0.304794:0.561024:0.304794:0.561024:0.295213:0.556420:0.295213:0.000000
6.2 Derivada de la función logarítmica:@0.073089:0.345370:0.371172:0.345370:0.371172:0.328423:0.073089:0.328423:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Demostremos ahora la derivada de la función logaritmo:  ( ) = :@0.073089:0.372732:0.532055:0.372732:0.532055:0.356298:0.073089:0.356298:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.490934:0.372732:0.507887:0.372732:0.507887:0.356326:0.490934:0.356326:0.000000:0.000000:0.000000
log  x:@0.532424:0.372732:0.570330:0.372733:0.570330:0.356327:0.532424:0.356326:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.554620:0.372478:0.559282:0.372478:0.559282:0.364217:0.554620:0.364217:0.000000
 o, si lo preferimos, :@0.570311:0.372733:0.710359:0.372733:0.710359:0.356299:0.570311:0.356299:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y  log  x:@0.073089:0.389375:0.135961:0.389375:0.135961:0.372969:0.073089:0.372969:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
= :@0.083743:0.389375:0.098671:0.389375:0.098671:0.372941:0.083743:0.372941:0.000000:0.000000
a:@0.120639:0.389120:0.125301:0.389120:0.125301:0.380859:0.120639:0.380859:0.000000
. :@0.135906:0.389375:0.143140:0.389375:0.143140:0.372941:0.135906:0.372941:0.000000:0.000000
Expresemos la igualdad en forma exponencial :@0.073101:0.419219:0.418651:0.419219:0.418651:0.402785:0.073101:0.402785:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a = x:@0.419396:0.419219:0.457598:0.419215:0.457598:0.402809:0.419396:0.402813:0.000000:0.000000:0.000000:0.000000:0.000000
y :@0.428674:0.413151:0.434984:0.413151:0.434984:0.403587:0.428674:0.403587:0.000000:0.000000
. Utilizaremos derivación implícita :@0.457617:0.419215:0.710284:0.419215:0.710284:0.402782:0.457617:0.402782:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
para derivar: :@0.073089:0.441904:0.166473:0.441904:0.166473:0.425470:0.073089:0.425470:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
, entonces :@0.257006:0.441909:0.335563:0.441909:0.335563:0.425475:0.257006:0.425475:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
.:@0.538244:0.441909:0.541447:0.441909:0.541447:0.425475:0.538244:0.425475:0.000000
Conclusión: si :@0.073089:0.480510:0.176328:0.480510:0.176328:0.464076:0.073089:0.464076:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f(x)= y = loga x.:@0.176151:0.480510:0.281846:0.480510:0.281846:0.464104:0.176151:0.464104:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
, entonces  '( )= :@0.281901:0.480510:0.403297:0.480510:0.403297:0.464076:0.281901:0.464076:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f  x:@0.360316:0.480510:0.383638:0.480510:0.383638:0.464104:0.360316:0.464104:0.000000:0.000000:0.000000:0.000000
.:@0.442925:0.480510:0.446127:0.480510:0.446127:0.464076:0.442925:0.464076:0.000000
Ahora bien, si tenemos  ( ) = :@0.073089:0.520004:0.283815:0.520004:0.283815:0.503571:0.073089:0.503571:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.243319:0.520004:0.260199:0.520004:0.260199:0.503598:0.243319:0.503598:0.000000:0.000000:0.000000
lnx:@0.283705:0.520004:0.304303:0.520004:0.304303:0.503598:0.283705:0.503598:0.000000:0.000000:0.000000
, entonces   '( ) = :@0.304247:0.520004:0.430300:0.520004:0.430300:0.503571:0.304247:0.503571:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.382662:0.520004:0.406684:0.520004:0.406684:0.503598:0.382662:0.503598:0.000000:0.000000:0.000000
.:@0.503208:0.520004:0.506411:0.520004:0.506411:0.503571:0.503208:0.503571:0.000000
Elaboraremos un cuadro de las fórmulas de derivadas de funciones exponenciales y :@0.073089:0.554515:0.710313:0.554515:0.710313:0.538081:0.073089:0.538081:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
logarítmicas elementales y compuestas: :@0.073089:0.571157:0.366385:0.571157:0.366385:0.554723:0.073089:0.554723:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Función exponencial  :@0.173069:0.606254:0.344017:0.606254:0.344017:0.589308:0.173069:0.589308:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
o logarítmica:@0.202760:0.622896:0.306743:0.622896:0.306743:0.605949:0.202760:0.605949:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Función derivada:@0.456467:0.614575:0.592953:0.614575:0.592953:0.597628:0.456467:0.597628:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.247968:0.649177:0.257227:0.649177:0.257227:0.632770:0.247968:0.632770:0.000000
x:@0.257159:0.643111:0.261505:0.643111:0.261505:0.633547:0.257159:0.633547:0.000000
a .lna:@0.505104:0.649175:0.544263:0.649175:0.544263:0.632769:0.505104:0.632769:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.514296:0.643111:0.518642:0.643111:0.518642:0.633547:0.514296:0.633547:0.000000
e:@0.248698:0.681252:0.256484:0.681252:0.256484:0.664846:0.248698:0.664846:0.000000
x:@0.256422:0.675182:0.260769:0.675182:0.260769:0.665618:0.256422:0.665618:0.000000
e:@0.518662:0.681246:0.526448:0.681246:0.526448:0.664840:0.518662:0.664840:0.000000
x:@0.526381:0.675182:0.530728:0.675182:0.530728:0.665618:0.526381:0.665618:0.000000
a:@0.242470:0.713318:0.251728:0.713318:0.251728:0.696912:0.242470:0.696912:0.000000
u x:@0.251661:0.707254:0.264196:0.707254:0.264196:0.697690:0.251661:0.697690:0.000000:0.000000:0.000000
( ):@0.257049:0.707254:0.266997:0.707254:0.266997:0.697673:0.257049:0.697673:0.000000:0.000000:0.000000
a x u x lna:@0.471966:0.713318:0.577371:0.713318:0.577371:0.696912:0.471966:0.696912:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
u:@0.481158:0.707254:0.486588:0.707254:0.486588:0.697690:0.481158:0.697690:0.000000
( ) .  '( ) . :@0.486550:0.713318:0.555080:0.713318:0.555080:0.696884:0.486550:0.696884:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
e:@0.243169:0.745395:0.250955:0.745395:0.250955:0.728989:0.243169:0.728989:0.000000
u x:@0.250925:0.739325:0.263460:0.739325:0.263460:0.729760:0.250925:0.729760:0.000000:0.000000:0.000000
( ):@0.256312:0.739325:0.266260:0.739325:0.266260:0.729744:0.256312:0.729744:0.000000:0.000000:0.000000
e u´(x):@0.496598:0.745389:0.552761:0.745389:0.552761:0.728983:0.496598:0.728983:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
u(x):@0.504318:0.739325:0.519653:0.739325:0.519653:0.729760:0.504318:0.729760:0.000000:0.000000:0.000000:0.000000
log  x:@0.236045:0.780808:0.273421:0.780811:0.273421:0.764405:0.236045:0.764402:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.258097:0.780556:0.262759:0.780556:0.262759:0.772295:0.258097:0.772295:0.000000
lnx:@0.244429:0.817491:0.265027:0.817491:0.265027:0.801085:0.244429:0.801085:0.000000:0.000000:0.000000
log u x:@0.221465:0.854264:0.278392:0.854264:0.278392:0.837858:0.221465:0.837858:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.243520:0.854009:0.248182:0.854009:0.248182:0.845749:0.243520:0.845749:0.000000
 [ ( )]:@0.248149:0.854264:0.288001:0.854264:0.288001:0.837831:0.248149:0.837831:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ln u x:@0.230204:0.893223:0.269665:0.893223:0.269665:0.876817:0.230204:0.876817:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
[ ( )]:@0.243361:0.893223:0.279274:0.893223:0.279274:0.876789:0.243361:0.876789:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Resolvamos algunos ejercicios de derivadas de funciones exponenciales y logarítmicas.:@0.073089:0.933710:0.703077:0.933710:0.703077:0.917276:0.073089:0.917276:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Competencia :@0.740294:0.145427:0.838985:0.145427:0.838985:0.130021:0.740294:0.130021:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
matemática :@0.740294:0.160556:0.828908:0.160556:0.828908:0.145150:0.740294:0.145150:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Observa el :@0.746659:0.186800:0.820466:0.186800:0.820466:0.171860:0.746659:0.171860:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
comportamiento :@0.746659:0.201928:0.864051:0.201928:0.864051:0.186989:0.746659:0.186989:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
alrededor de cero: la :@0.746659:0.217057:0.884650:0.217057:0.884650:0.202118:0.746659:0.202118:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
clave es ver cómo el :@0.746659:0.232186:0.883256:0.232186:0.883256:0.217247:0.746659:0.217247:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
cambio en   alrededor :@0.746659:0.247315:0.903912:0.247315:0.903912:0.232375:0.746659:0.232375:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.820740:0.247315:0.829157:0.247315:0.829157:0.232401:0.820740:0.232401:0.000000
x:@0.829100:0.241803:0.833051:0.241803:0.833051:0.233107:0.829100:0.233107:0.000000
de 0 resulta en una :@0.746653:0.262444:0.875999:0.262444:0.875999:0.247504:0.746653:0.247504:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
constante multiplicativa, :@0.746653:0.277573:0.911759:0.277573:0.911759:0.262633:0.746653:0.262633:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ln(a):@0.746653:0.292702:0.775753:0.292702:0.775753:0.277788:0.746653:0.277788:0.000000:0.000000:0.000000:0.000000:0.000000
. Esto indica la :@0.775719:0.292702:0.871564:0.292702:0.871564:0.277762:0.775719:0.277762:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
velocidad de cambio de :@0.746653:0.307831:0.909517:0.307831:0.909517:0.292891:0.746653:0.292891:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.746653:0.322960:0.755070:0.322960:0.755070:0.308045:0.746653:0.308045:0.000000
x:@0.755016:0.317447:0.759201:0.317447:0.759201:0.308737:0.755016:0.308737:0.000000
 y se convierte en  '( )= :@0.759159:0.322960:0.913981:0.322960:0.913981:0.308020:0.759159:0.308020:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
f x:@0.877803:0.322960:0.896043:0.322960:0.896043:0.308045:0.877803:0.308045:0.000000:0.000000:0.000000
a ln(a).:@0.746643:0.338089:0.794819:0.338089:0.794819:0.323174:0.746643:0.323174:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.755016:0.332576:0.759201:0.332576:0.759201:0.323866:0.755016:0.323866:0.000000
 :@0.759159:0.338089:0.762824:0.338089:0.762824:0.323149:0.759159:0.323149:0.000000
Competencia :@0.740294:0.393209:0.838985:0.393209:0.838985:0.377802:0.740294:0.377802:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
matemática :@0.740294:0.408338:0.828908:0.408338:0.828908:0.392931:0.740294:0.392931:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
La derivada :@0.746659:0.434580:0.825971:0.434580:0.825971:0.419640:0.746659:0.419640:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
lnx:@0.825836:0.434580:0.844561:0.434580:0.844561:0.419665:0.825836:0.419665:0.000000:0.000000:0.000000
 solo es :@0.844511:0.434580:0.896988:0.434580:0.896988:0.419640:0.844511:0.419640:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
válida para  > 0. Esto :@0.746651:0.449709:0.890555:0.449709:0.890555:0.434769:0.746651:0.434769:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x :@0.821368:0.449709:0.831090:0.449709:0.831090:0.434794:0.821368:0.434794:0.000000:0.000000
significa que la función :@0.746651:0.464838:0.902169:0.464838:0.902169:0.449898:0.746651:0.449898:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
logarítmica natural solo :@0.746651:0.479967:0.905532:0.479967:0.905532:0.465027:0.746651:0.465027:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
está definida en el :@0.746651:0.495096:0.870247:0.495096:0.870247:0.480156:0.746651:0.480156:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
dominio positivo de los :@0.746651:0.510225:0.905214:0.510225:0.905214:0.495285:0.746651:0.495285:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
números reales, lo cual es :@0.746651:0.525354:0.918253:0.525354:0.918253:0.510414:0.746651:0.510414:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
crucial a la hora de aplicar :@0.746651:0.540483:0.920466:0.540483:0.920466:0.525543:0.746651:0.525543:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
la derivada en contextos :@0.746651:0.555612:0.911737:0.555612:0.911737:0.540672:0.746651:0.540672:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
prácticos.:@0.746651:0.570741:0.809460:0.570741:0.809460:0.555801:0.746651:0.555801:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000