134:@0.031627:0.974518:0.062330:0.974518:0.062330:0.956333:0.031627:0.956333:0.000000:0.000000:0.000000 Derivadas:@0.073089:0.043661:0.163191:0.043661:0.163191:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 La regla de la cadena de derivadas es una herramienta muy valiosa en el cálculo, ya que :@0.293660:0.241667:0.931446:0.241667:0.931446:0.225233:0.293660:0.225233:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 permite encontrar la derivada de una función que está compuesta por otras funciones, :@0.293660:0.258309:0.931079:0.258309:0.931079:0.241875:0.293660:0.241875:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de tal forma que se puede encontrar la derivada de funciones mucho más complejas. :@0.293660:0.274951:0.930415:0.274951:0.930415:0.258517:0.293660:0.258517:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Esta regla es útil en el cálculo de derivadas de ciertas funciones trigonométricas, :@0.293660:0.291593:0.931230:0.291593:0.931230:0.275159:0.293660:0.275159:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 exponenciales y logarítmicas, como también en la solución de problemas de :@0.293660:0.308235:0.930947:0.308235:0.930947:0.291801:0.293660:0.291801:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 optimización y en el análisis de problemas del área de las ciencias y la ingenierías.:@0.293660:0.324876:0.882019:0.324876:0.882019:0.308443:0.293660:0.308443:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Recordemos que :@0.293660:0.352238:0.421576:0.352238:0.421576:0.335804:0.293660:0.335804:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (f g) (x) = f (g(x)).:@0.421406:0.352238:0.542029:0.352238:0.542029:0.335832:0.421406:0.335832:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 º:@0.434089:0.354762:0.440292:0.354762:0.440292:0.338328:0.434089:0.338328:0.000000 Si consideramos la función = ( ( )), haciendo = ( ), tendremos que = ( ). Si es :@0.293660:0.384399:0.930993:0.384399:0.930993:0.367965:0.293660:0.367965:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.494631:0.384399:0.502104:0.384399:0.502104:0.367993:0.494631:0.367993:0.000000 f g x:@0.521855:0.384399:0.552945:0.384399:0.552945:0.367993:0.521855:0.367993:0.000000:0.000000:0.000000:0.000000:0.000000 u g x:@0.643324:0.384399:0.693153:0.384399:0.693153:0.367993:0.643324:0.367993:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y f u:@0.823071:0.384399:0.868353:0.384399:0.868353:0.367993:0.823071:0.367993:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 g:@0.897528:0.384399:0.906787:0.384399:0.906787:0.367993:0.897528:0.367993:0.000000 derivable en y derivable en , entonces :@0.293697:0.405825:0.610353:0.405825:0.610353:0.389391:0.293697:0.389391:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x f:@0.390133:0.405825:0.419695:0.405825:0.419695:0.389419:0.390133:0.389419:0.000000:0.000000:0.000000 u:@0.520787:0.405825:0.530101:0.405825:0.530101:0.389419:0.520787:0.389419:0.000000 dy:@0.616886:0.398987:0.633545:0.398987:0.633545:0.382581:0.616886:0.382581:0.000000:0.000000 dx:@0.616905:0.415379:0.633545:0.415379:0.633545:0.398973:0.616905:0.398973:0.000000:0.000000 = ´( ( )) ´( ), que lo podemos escribir :@0.639583:0.405824:0.931039:0.405824:0.931039:0.389390:0.639583:0.389390:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f g x g x:@0.659849:0.405824:0.733736:0.405824:0.733736:0.389418:0.659849:0.389418:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 como sigue::@0.293655:0.427250:0.382452:0.427250:0.382452:0.410816:0.293655:0.410816:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (:@0.293655:0.452462:0.298533:0.452462:0.298533:0.436029:0.293655:0.436029:0.000000 f g x:@0.298533:0.452462:0.349447:0.452462:0.349447:0.436056:0.298533:0.436056:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 º :@0.306853:0.454986:0.317088:0.454986:0.317088:0.438553:0.306853:0.438553:0.000000:0.000000 )´( ) = ´( ( )) ´( ) o lo que es lo mismo :@0.326733:0.452462:0.619841:0.452462:0.619841:0.436029:0.326733:0.436029:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f g x g x:@0.374132:0.452462:0.447613:0.452462:0.447613:0.436056:0.374132:0.436056:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 d f g(x:@0.633389:0.446528:0.678248:0.446528:0.678248:0.430122:0.633389:0.430122:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 [ (:@0.642575:0.446528:0.656877:0.446528:0.656877:0.430094:0.642575:0.430094:0.000000:0.000000:0.000000 ))]:@0.678193:0.446528:0.692679:0.446528:0.692679:0.430094:0.678193:0.430094:0.000000:0.000000:0.000000 dx:@0.654705:0.462920:0.671345:0.462920:0.671345:0.446514:0.654705:0.446514:0.000000:0.000000 = ´( ( )) ´( ), que también se :@0.705951:0.452478:0.931044:0.452478:0.931044:0.436044:0.705951:0.436044:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f g x g x:@0.725775:0.452478:0.799257:0.452478:0.799257:0.436072:0.725775:0.436072:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 escribe como :@0.293665:0.477690:0.396206:0.477690:0.396206:0.461256:0.293665:0.461256:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.402032:0.469957:0.418691:0.469957:0.418691:0.453550:0.402032:0.453550:0.000000:0.000000 dx:@0.402050:0.486349:0.418691:0.486349:0.418691:0.469943:0.402050:0.469943:0.000000:0.000000 = :@0.424618:0.477695:0.443504:0.477695:0.443504:0.461261:0.424618:0.461261:0.000000:0.000000:0.000000 dy:@0.449389:0.469957:0.466047:0.469957:0.466047:0.453550:0.449389:0.453550:0.000000:0.000000 du:@0.448468:0.486349:0.466968:0.486349:0.466968:0.469943:0.448468:0.469943:0.000000:0.000000 :@0.471956:0.477695:0.487050:0.477689:0.487050:0.461255:0.471956:0.461261:0.000000:0.000000:0.000000 x:@0.475914:0.477374:0.483093:0.477374:0.483093:0.462434:0.475914:0.462434:0.000000 du:@0.492040:0.470849:0.510539:0.470849:0.510539:0.454443:0.492040:0.454443:0.000000:0.000000 dx:@0.492960:0.487241:0.509601:0.487241:0.509601:0.470835:0.492960:0.470835:0.000000:0.000000 .:@0.515528:0.477686:0.518731:0.477686:0.518731:0.461252:0.515528:0.461252:0.000000 Ejemplo 1:@0.293647:0.520040:0.369429:0.520040:0.369429:0.503370:0.293647:0.503370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . Si = (2 + 1) , encuentra :@0.369319:0.520040:0.566292:0.520037:0.566292:0.503603:0.369319:0.503606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.396525:0.520040:0.403998:0.520040:0.403998:0.503634:0.396525:0.503634:0.000000 x :@0.436413:0.520040:0.447108:0.520040:0.447108:0.503634:0.436413:0.503634:0.000000:0.000000 3:@0.475601:0.513973:0.480838:0.513973:0.480838:0.504392:0.475601:0.504392:0.000000 dy:@0.572123:0.512305:0.588782:0.512305:0.588782:0.495899:0.572123:0.495899:0.000000:0.000000 d:@0.571755:0.528697:0.581769:0.528697:0.581769:0.512263:0.571755:0.512263:0.000000 x:@0.581695:0.528697:0.589150:0.528697:0.589150:0.512291:0.581695:0.512291:0.000000 . :@0.594691:0.520043:0.601925:0.520043:0.601925:0.503610:0.594691:0.503610:0.000000:0.000000 Haciendo = 2x+1 tendremos que = . Ahora, podemos determinar :@0.293658:0.548099:0.808841:0.548091:0.808841:0.531657:0.293658:0.531665:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.367582:0.548099:0.376896:0.548099:0.376896:0.531693:0.367582:0.531693:0.000000 y u:@0.551949:0.548099:0.586757:0.548099:0.586757:0.531693:0.551949:0.531693:0.000000:0.000000:0.000000:0.000000 3:@0.586700:0.542027:0.591937:0.542027:0.591937:0.532446:0.586700:0.532446:0.000000 dy:@0.814572:0.540359:0.831231:0.540359:0.831231:0.523953:0.814572:0.523953:0.000000:0.000000 du:@0.813652:0.556751:0.832151:0.556751:0.832151:0.540345:0.813652:0.540345:0.000000:0.000000 = :@0.837139:0.548098:0.856025:0.548098:0.856025:0.531664:0.837139:0.531664:0.000000:0.000000:0.000000 du:@0.861008:0.541253:0.879508:0.541253:0.879508:0.524847:0.861008:0.524847:0.000000:0.000000 dx:@0.861929:0.557645:0.878569:0.557645:0.878569:0.541239:0.861929:0.541239:0.000000:0.000000 .:@0.884496:0.548090:0.891657:0.548090:0.891657:0.531656:0.884496:0.531656:0.000000:0.000000 :@0.293660:0.580611:0.361012:0.580611:0.361012:0.564177:0.293660:0.564177:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.366807:0.571987:0.383466:0.571987:0.383466:0.555581:0.366807:0.555581:0.000000:0.000000 du:@0.365887:0.588380:0.384386:0.588380:0.384386:0.571974:0.365887:0.571974:0.000000:0.000000 = 3 :@0.389374:0.580613:0.443460:0.580612:0.443460:0.564178:0.389374:0.564180:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.417059:0.580613:0.426373:0.580613:0.426373:0.564207:0.417059:0.564207:0.000000 2:@0.426323:0.574548:0.431560:0.574548:0.431560:0.564967:0.426323:0.564967:0.000000 du:@0.448433:0.573774:0.466933:0.573774:0.466933:0.557368:0.448433:0.557368:0.000000:0.000000 dx:@0.449354:0.590166:0.465994:0.590166:0.465994:0.573760:0.449354:0.573760:0.000000:0.000000 = 2, por tanto, :@0.471921:0.580611:0.581579:0.580611:0.581579:0.564177:0.471921:0.564177:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.587395:0.573774:0.604053:0.573774:0.604053:0.557368:0.587395:0.557368:0.000000:0.000000 dx:@0.587395:0.590166:0.604035:0.590166:0.604035:0.573760:0.587395:0.573760:0.000000:0.000000 = 3 (2) = 6:@0.609962:0.580611:0.702371:0.580612:0.702371:0.564178:0.609962:0.564177:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.637647:0.580611:0.646961:0.580611:0.646961:0.564205:0.637647:0.564205:0.000000 2:@0.646910:0.574548:0.652147:0.574548:0.652147:0.564967:0.646910:0.564967:0.000000 u:@0.702243:0.580612:0.711557:0.580612:0.711557:0.564206:0.702243:0.564206:0.000000 2:@0.711488:0.574548:0.716725:0.574548:0.716725:0.564967:0.711488:0.564967:0.000000 :@0.293660:0.612240:0.396630:0.612240:0.396630:0.595807:0.293660:0.595807:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.402362:0.604508:0.419020:0.604508:0.419020:0.588102:0.402362:0.588102:0.000000:0.000000 dx:@0.402362:0.620901:0.419002:0.620901:0.419002:0.604494:0.402362:0.604494:0.000000:0.000000 = 6(2x + 1) .:@0.424929:0.612247:0.515027:0.612240:0.515027:0.595807:0.424929:0.595813:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.506635:0.606177:0.511872:0.606177:0.511872:0.596596:0.506635:0.596596:0.000000 Ejemplo 2.:@0.293662:0.647438:0.373181:0.647438:0.373181:0.630768:0.293662:0.630768:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Si = :@0.373071:0.647438:0.419475:0.647438:0.419475:0.631004:0.373071:0.631004:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.393190:0.647438:0.400663:0.647438:0.400663:0.631032:0.393190:0.631032:0.000000 cos x:@0.419383:0.647438:0.462548:0.647438:0.462548:0.631032:0.419383:0.631032:0.000000:0.000000:0.000000:0.000000:0.000000 5 , encuentra :@0.442245:0.647438:0.547976:0.647438:0.547976:0.631004:0.442245:0.631004:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.553828:0.639710:0.570487:0.639710:0.570487:0.623304:0.553828:0.623304:0.000000:0.000000 dx:@0.553828:0.656102:0.570468:0.656102:0.570468:0.639696:0.553828:0.639696:0.000000:0.000000 . :@0.576396:0.647448:0.583630:0.647448:0.583630:0.631014:0.576396:0.631014:0.000000:0.000000 Haciendo = 5 , tendremos que = s . Ahora, podemos determinar :@0.293660:0.674602:0.817792:0.674602:0.817792:0.658168:0.293660:0.658168:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.367584:0.674602:0.376898:0.674602:0.376898:0.658196:0.367584:0.658196:0.000000 x:@0.404509:0.674602:0.411964:0.674602:0.411964:0.658196:0.404509:0.658196:0.000000 y co u:@0.534850:0.674602:0.596956:0.674602:0.596956:0.658196:0.534850:0.658196:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.823543:0.667764:0.840202:0.667764:0.840202:0.651358:0.823543:0.651358:0.000000:0.000000 du:@0.822623:0.684156:0.841122:0.684156:0.841122:0.667750:0.822623:0.667750:0.000000:0.000000 y :@0.846110:0.674601:0.862125:0.674601:0.862125:0.658167:0.846110:0.658167:0.000000:0.000000:0.000000 du:@0.867108:0.668658:0.885607:0.668658:0.885607:0.652251:0.867108:0.652251:0.000000:0.000000 dx:@0.868028:0.685050:0.884669:0.685050:0.884669:0.668644:0.868028:0.668644:0.000000:0.000000 .:@0.890596:0.674607:0.893799:0.674607:0.893799:0.658173:0.890596:0.658173:0.000000 :@0.293648:0.699084:0.364958:0.699084:0.364958:0.682651:0.293648:0.682651:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.370760:0.691351:0.387418:0.691351:0.387418:0.674945:0.370760:0.674945:0.000000:0.000000 du:@0.369839:0.707743:0.388339:0.707743:0.388339:0.691337:0.369839:0.691337:0.000000:0.000000 = –:@0.393327:0.699089:0.421343:0.699089:0.421343:0.682656:0.393327:0.682656:0.000000:0.000000:0.000000:0.000000 sen u:@0.421232:0.699089:0.457219:0.699089:0.457219:0.682683:0.421232:0.682683:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.457182:0.699089:0.469128:0.699089:0.469128:0.682656:0.457182:0.682656:0.000000:0.000000:0.000000 du:@0.474120:0.691351:0.492619:0.691351:0.492619:0.674945:0.474120:0.674945:0.000000:0.000000 dx:@0.475040:0.707743:0.491680:0.707743:0.491680:0.691337:0.475040:0.691337:0.000000:0.000000 = 5, por tanto::@0.497607:0.699089:0.603644:0.699089:0.603644:0.682656:0.497607:0.682656:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = (–:@0.293655:0.719309:0.409584:0.719309:0.409584:0.702875:0.293655:0.702875:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 sen u:@0.409326:0.719309:0.446049:0.719309:0.446049:0.702903:0.409326:0.702903:0.000000:0.000000:0.000000:0.000000:0.000000 ) (5):@0.432777:0.719309:0.473328:0.719309:0.473328:0.702875:0.432777:0.702875:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = –5:@0.293655:0.739529:0.417646:0.739529:0.417646:0.723095:0.293655:0.723095:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 sen u:@0.417389:0.739529:0.454111:0.739529:0.454111:0.723123:0.417389:0.723123:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.440840:0.739529:0.444871:0.739529:0.444871:0.723095:0.440840:0.723095:0.000000 = –5:@0.293674:0.759749:0.417665:0.759749:0.417665:0.743315:0.293674:0.743315:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 sen:@0.417407:0.759749:0.440913:0.759749:0.440913:0.743343:0.417407:0.743343:0.000000:0.000000:0.000000 5x. :@0.440858:0.759749:0.674365:0.759749:0.674365:0.743315:0.440858:0.743315:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo 3. :@0.293674:0.809175:0.377022:0.809175:0.377022:0.792506:0.293674:0.792506:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Determina :@0.376893:0.809175:0.462354:0.809175:0.462354:0.792741:0.376893:0.792741:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.554797:0.809160:0.558000:0.809160:0.558000:0.792726:0.554797:0.792726:0.000000 Solución::@0.293660:0.836522:0.360085:0.836522:0.360085:0.820088:0.293660:0.820088:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Tema 4. Regla de la cadena:@0.073089:0.098522:0.470180:0.098522:0.470180:0.067281:0.073089:0.067281:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Si la velocidad de un ciclista al bajar una montaña depende de la inclinación de la pendiente :@0.293660:0.166113:0.907378:0.166113:0.907378:0.151173:0.293660:0.151173:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 en cada punto de su recorrido, y la inclinación cambia en función de la distancia que ha :@0.293660:0.181242:0.907000:0.181242:0.907000:0.166302:0.293660:0.166302:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 recorrido, ¿cómo podríamos expresar el cambio de la inclinación en función del tiempo?:@0.293660:0.196371:0.875860:0.196371:0.875860:0.181431:0.293660:0.181431:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Reto matemático:@0.302768:0.139600:0.437491:0.139600:0.437491:0.122653:0.302768:0.122653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Competencia :@0.078581:0.245492:0.177272:0.245492:0.177272:0.230086:0.078581:0.230086:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemática :@0.078581:0.260621:0.167195:0.260621:0.167195:0.245215:0.078581:0.245215:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Identificar claramente las :@0.080995:0.288746:0.257267:0.288746:0.257267:0.273592:0.080995:0.273592:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 variables intermedias. :@0.080995:0.305136:0.238209:0.305136:0.238209:0.289981:0.080995:0.289981:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Si tienes una variable que :@0.080995:0.321525:0.252743:0.321525:0.252743:0.306585:0.080995:0.306585:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 depende de otra, y esta :@0.080995:0.337915:0.239832:0.337915:0.239832:0.322975:0.080995:0.322975:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 última depende de una :@0.080995:0.354305:0.240150:0.354305:0.240150:0.339365:0.080995:0.339365:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 tercera, entonces estás en :@0.080995:0.370694:0.255223:0.370694:0.255223:0.355755:0.080995:0.355755:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 un caso ideal para aplicar :@0.080995:0.387084:0.250810:0.387084:0.250810:0.372144:0.080995:0.372144:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la regla de la cadena.:@0.080995:0.403474:0.218626:0.403474:0.218626:0.388534:0.080995:0.388534:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Por ejemplo, en una :@0.080995:0.419863:0.216592:0.419863:0.216592:0.404924:0.080995:0.404924:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 situación donde y :@0.080995:0.436253:0.202700:0.436253:0.202700:0.421313:0.080995:0.421313:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 depende de , y :@0.080995:0.452643:0.203336:0.452643:0.203336:0.437703:0.080995:0.437703:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.165501:0.452643:0.173968:0.452643:0.173968:0.437728:0.165501:0.437728:0.000000 u:@0.191204:0.452643:0.199671:0.452643:0.199671:0.437728:0.191204:0.437728:0.000000 depende de x (como :@0.080978:0.469032:0.222990:0.469032:0.222990:0.454093:0.080978:0.454093:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y u x:@0.080978:0.485422:0.111618:0.485422:0.111618:0.470508:0.080978:0.470508:0.000000:0.000000:0.000000:0.000000:0.000000 ( ( ))), primero calcula :@0.087705:0.485422:0.236840:0.485422:0.236840:0.470482:0.087705:0.470482:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la derivada de y con :@0.080978:0.504333:0.216656:0.504333:0.216656:0.489393:0.080978:0.489393:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 respecto a :@0.080978:0.523244:0.165250:0.523244:0.165250:0.508305:0.080978:0.508305:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.153184:0.523244:0.161652:0.523244:0.161652:0.508330:0.153184:0.508330:0.000000 dy:@0.169674:0.516704:0.184818:0.516704:0.184818:0.501789:0.169674:0.501789:0.000000:0.000000 du:@0.168837:0.533093:0.185655:0.533093:0.185655:0.518179:0.168837:0.518179:0.000000:0.000000 y luego :@0.189236:0.523247:0.242081:0.523247:0.242081:0.508307:0.189236:0.508307:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la derivada de con :@0.081001:0.542158:0.217114:0.542158:0.217114:0.527218:0.081001:0.527218:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u :@0.176886:0.542158:0.188299:0.542158:0.188299:0.527244:0.176886:0.527244:0.000000:0.000000 respecto a :@0.081001:0.558548:0.163583:0.558548:0.163583:0.543608:0.081001:0.543608:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x (du/dx:@0.153208:0.558548:0.204939:0.558548:0.204939:0.543633:0.153208:0.543633:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ). Al :@0.204899:0.558548:0.236241:0.558548:0.236241:0.543608:0.204899:0.543608:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 final, multiplica ambos :@0.081001:0.574938:0.233072:0.574938:0.233072:0.559998:0.081001:0.559998:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 resultados::@0.081001:0.591327:0.151795:0.591327:0.151795:0.576387:0.081001:0.576387:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.085473:0.607822:0.100618:0.607822:0.100618:0.592907:0.085473:0.592907:0.000000:0.000000 dx:@0.085490:0.624211:0.100618:0.624211:0.100618:0.609297:0.085490:0.609297:0.000000:0.000000 = :@0.105035:0.616155:0.122204:0.616155:0.122204:0.601215:0.105035:0.601215:0.000000:0.000000:0.000000 dy:@0.126591:0.608714:0.141735:0.608714:0.141735:0.593800:0.126591:0.593800:0.000000:0.000000 du:@0.125755:0.625104:0.142572:0.625104:0.142572:0.610189:0.125755:0.610189:0.000000:0.000000 x :@0.146153:0.616153:0.160527:0.616153:0.160527:0.601213:0.146153:0.601213:0.000000:0.000000:0.000000 du:@0.164078:0.608714:0.180896:0.608714:0.180896:0.593800:0.164078:0.593800:0.000000:0.000000 dx:@0.164915:0.625104:0.180042:0.625104:0.180042:0.610189:0.164915:0.610189:0.000000:0.000000 Función exterior :@0.115792:0.678476:0.214572:0.678476:0.214572:0.665713:0.115792:0.665713:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.214572:0.678476:0.218687:0.678476:0.218687:0.665713:0.214572:0.665713:0.000000 Función interna :@0.118808:0.731364:0.213894:0.731364:0.213894:0.718601:0.118808:0.718601:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 g:@0.213894:0.731364:0.221772:0.731364:0.221772:0.718601:0.213894:0.718601:0.000000 y:@0.096492:0.707993:0.105496:0.707993:0.105496:0.689487:0.096492:0.689487:0.000000 = (5x 6):@0.105496:0.707993:0.187189:0.707993:0.187189:0.689503:0.105496:0.689503:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 _:@0.156378:0.701840:0.166587:0.701840:0.166587:0.683349:0.156378:0.683349:0.000000 4:@0.187189:0.700686:0.194521:0.700686:0.194521:0.687742:0.187189:0.687742:0.000000 = ‘ :@0.146370:0.760718:0.175571:0.760718:0.175571:0.744370:0.146370:0.744370:0.000000:0.000000:0.000000:0.000000 f :@0.157120:0.760718:0.168790:0.760718:0.168790:0.744370:0.157120:0.744370:0.000000:0.000000:0.000000 ( ( )) ( ):@0.175571:0.760718:0.247663:0.760718:0.247663:0.744384:0.175571:0.744384:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 g:@0.180694:0.760718:0.190145:0.760718:0.190145:0.744370:0.180694:0.744370:0.000000 x:@0.195267:0.760718:0.203221:0.760718:0.203221:0.744370:0.195267:0.744370:0.000000 g:@0.213466:0.760718:0.222917:0.760718:0.222917:0.744370:0.213466:0.744370:0.000000 ’ x:@0.222682:0.760718:0.242541:0.760718:0.242541:0.744370:0.222682:0.744370:0.000000:0.000000:0.000000:0.000000 = 20(5 6):@0.146370:0.776603:0.232819:0.776603:0.232819:0.760269:0.146370:0.760269:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.193824:0.776603:0.201778:0.776603:0.201778:0.760256:0.193824:0.760256:0.000000 _:@0.205602:0.771290:0.214620:0.771290:0.214620:0.754956:0.205602:0.754956:0.000000 3:@0.232817:0.770110:0.238842:0.770110:0.238842:0.759475:0.232817:0.759475:0.000000 d :@0.092706:0.751953:0.112295:0.751953:0.112295:0.741309:0.092706:0.741309:0.000000:0.000000:0.000000:0.000000 [ ( ( ))]:@0.101009:0.751953:0.140306:0.751953:0.140306:0.741318:0.101009:0.741318:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f:@0.106834:0.751953:0.110158:0.751953:0.110158:0.741309:0.106834:0.741309:0.000000 g:@0.115631:0.751953:0.121785:0.751953:0.121785:0.741309:0.115631:0.741309:0.000000 x:@0.125120:0.751953:0.130300:0.751953:0.130300:0.741309:0.125120:0.741309:0.000000 dx:@0.110158:0.764111:0.121503:0.764111:0.121503:0.753467:0.110158:0.753467:0.000000:0.000000 20:@0.489328:0.911770:0.507219:0.911770:0.507219:0.895336:0.489328:0.895336:0.000000:0.000000 DBA 8 (Ev. 2) Relaciona características algebraicas de las funciones, sus gráficas y procesos de aproximación sucesiva.:@0.305749:0.971366:0.916832:0.971366:0.916832:0.959414:0.305749:0.959414:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 MUESTRA EDITORIAL:@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