131:@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 Utilizando la forma punto–pendiente de la recta, encontramos que la :@0.073089:0.136836:0.618124:0.136836:0.618124:0.120402:0.073089:0.120402:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ecuación de la recta :@0.073089:0.153478:0.222523:0.153478:0.222523:0.137044:0.073089:0.137044:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 tan:@0.222316:0.153478:0.246301:0.153478:0.246301:0.137072:0.222316:0.137072:0.000000:0.000000:0.000000 gente es la que sigue::@0.246246:0.153478:0.403494:0.153478:0.403494:0.137044:0.246246:0.137044:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 – 0 = –2( – ) = –2 + 2 . :@0.182722:0.173698:0.512415:0.173698:0.512415:0.157264:0.182722:0.157264:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.269642:0.173698:0.277115:0.173698:0.277115:0.157292:0.269642:0.157292:0.000000 x:@0.344577:0.173698:0.352032:0.173698:0.352032:0.157292:0.344577:0.157292:0.000000 p:@0.361089:0.173143:0.373366:0.173143:0.373366:0.159428:0.361089:0.159428:0.000000 y:@0.409684:0.173698:0.417157:0.173698:0.417157:0.157292:0.409684:0.157292:0.000000 x:@0.453898:0.173698:0.461353:0.173698:0.461353:0.157292:0.453898:0.157292:0.000000 p:@0.488964:0.173143:0.501242:0.173143:0.501242:0.159428:0.488964:0.159428:0.000000 Gráficamente, y con el uso de un aplicativo web o una calculadora gráfica, :@0.073052:0.201060:0.617955:0.201060:0.617955:0.184626:0.073052:0.184626:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 podemos constatar la validez del resultado.:@0.073052:0.217702:0.386294:0.217702:0.386294:0.201268:0.073052:0.201268:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3.3 Derivada de la función tangente:@0.073052:0.341448:0.353348:0.341448:0.353348:0.324501:0.073052:0.324501:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Sea ( ) = :@0.073052:0.368810:0.143165:0.368810:0.143165:0.352376:0.073052:0.352376:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.102669:0.368810:0.119549:0.368810:0.119549:0.352403:0.102669:0.352403:0.000000:0.000000:0.000000 tan x:@0.143055:0.368810:0.177660:0.368810:0.177660:0.352403:0.143055:0.352403:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.177624:0.368810:0.180827:0.368810:0.180827:0.352376:0.177624:0.352376:0.000000 Como :@0.073034:0.411898:0.121305:0.411898:0.121305:0.395464:0.073034:0.395464:0.000000:0.000000:0.000000:0.000000:0.000000 tan x:@0.120561:0.411898:0.154338:0.411898:0.154338:0.395492:0.120561:0.395492:0.000000:0.000000:0.000000:0.000000:0.000000 = :@0.154191:0.411898:0.172258:0.411898:0.172258:0.395464:0.154191:0.395464:0.000000:0.000000:0.000000 sen x:@0.189977:0.404341:0.224104:0.404341:0.224104:0.387935:0.189977:0.387935:0.000000:0.000000:0.000000:0.000000:0.000000 cos x:@0.190272:0.420733:0.223817:0.420733:0.223817:0.404327:0.190272:0.404327:0.000000:0.000000:0.000000:0.000000:0.000000 , entonces podemos aplicar la regla de la derivada de un cociente::@0.242346:0.411899:0.705934:0.411899:0.705934:0.395465:0.242346:0.395465:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.073091:0.451923:0.095401:0.451923:0.095401:0.435517:0.073091:0.435517:0.000000:0.000000:0.000000 ´( )= :@0.077711:0.451923:0.115060:0.451923:0.115060:0.435489:0.077711:0.435489:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cos x cos x + sen x sen x:@0.122615:0.443651:0.281595:0.443651:0.281595:0.427245:0.122615:0.427245:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cos x:@0.184427:0.460043:0.219941:0.460041:0.219941:0.443635:0.184427:0.443637:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.207297:0.453977:0.212533:0.453977:0.212533:0.444396:0.207297:0.444396:0.000000 = :@0.289318:0.451914:0.308204:0.451914:0.308204:0.435480:0.289318:0.435480:0.000000:0.000000:0.000000 cos x + sen x:@0.310796:0.442936:0.406224:0.442936:0.406224:0.426530:0.310796:0.426530:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.333661:0.436872:0.338898:0.436872:0.338898:0.427291:0.333661:0.427291:0.000000 2:@0.390338:0.436872:0.395575:0.436872:0.395575:0.427291:0.390338:0.427291:0.000000 cos x:@0.339129:0.459328:0.377885:0.459326:0.377885:0.442920:0.339129:0.442922:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.362000:0.453262:0.367236:0.453262:0.367236:0.443681:0.362000:0.443681:0.000000 = :@0.408846:0.451920:0.427732:0.451920:0.427732:0.435487:0.408846:0.435487:0.000000:0.000000:0.000000 1:@0.446994:0.443651:0.455977:0.443651:0.455977:0.427217:0.446994:0.427217:0.000000 cos x:@0.433723:0.460043:0.469240:0.460041:0.469240:0.443635:0.433723:0.443637:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.456596:0.453977:0.461833:0.453977:0.461833:0.444396:0.456596:0.444396:0.000000 = :@0.475259:0.451914:0.494145:0.451914:0.494145:0.435480:0.475259:0.435480:0.000000:0.000000:0.000000 sec x:@0.494053:0.451914:0.532262:0.451919:0.532262:0.435513:0.494053:0.435508:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.515660:0.445855:0.520897:0.445855:0.520897:0.436274:0.515660:0.436274:0.000000 :@0.520849:0.451919:0.524880:0.451919:0.524880:0.435485:0.520849:0.435485:0.000000 Recordando que :@0.073093:0.492484:0.198376:0.492484:0.198376:0.476050:0.073093:0.476050:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cot x :@0.198207:0.492484:0.233998:0.492484:0.233998:0.476077:0.198207:0.476077:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = :@0.233954:0.492484:0.248882:0.492484:0.248882:0.476050:0.233954:0.476050:0.000000:0.000000 cos x:@0.254995:0.485643:0.288540:0.485643:0.288540:0.469237:0.254995:0.469237:0.000000:0.000000:0.000000:0.000000:0.000000 senx:@0.256320:0.502035:0.287208:0.502035:0.287208:0.485629:0.256320:0.485629:0.000000:0.000000:0.000000:0.000000 , :@0.294662:0.492480:0.301823:0.492480:0.301823:0.476046:0.294662:0.476046:0.000000:0.000000 sec x:@0.301731:0.492480:0.334017:0.492480:0.334017:0.476074:0.301731:0.476074:0.000000:0.000000:0.000000:0.000000:0.000000 = :@0.333980:0.492480:0.348909:0.492480:0.348909:0.476046:0.333980:0.476046:0.000000:0.000000 1:@0.376101:0.483856:0.385084:0.483856:0.385084:0.467422:0.376101:0.467422:0.000000 cos x:@0.363805:0.500248:0.397351:0.500248:0.397351:0.483842:0.363805:0.483842:0.000000:0.000000:0.000000:0.000000:0.000000 , y que :@0.412271:0.492482:0.467788:0.492482:0.467788:0.476048:0.412271:0.476048:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 csc x :@0.467659:0.492482:0.503056:0.492482:0.503056:0.476076:0.467659:0.476076:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = :@0.503019:0.492482:0.517947:0.492482:0.517947:0.476048:0.503019:0.476048:0.000000:0.000000 1:@0.535319:0.484214:0.544302:0.484214:0.544302:0.467781:0.535319:0.467781:0.000000 sen x:@0.522728:0.500607:0.556855:0.500607:0.556855:0.484200:0.522728:0.484200:0.000000:0.000000:0.000000:0.000000:0.000000 , resulta sencillo :@0.561678:0.492480:0.679774:0.492480:0.679774:0.476046:0.561678:0.476046:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 deducir las derivadas de las funciones trigonométricas: :@0.073076:0.515168:0.476894:0.515168:0.476894:0.498734:0.073076:0.498734:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y cot x y sec x y csc x:@0.073076:0.537857:0.270291:0.537857:0.270291:0.521450:0.073076:0.521450:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = :@0.083734:0.537857:0.098662:0.537857:0.098662:0.521423:0.083734:0.521423:0.000000:0.000000 , = :@0.131077:0.537857:0.163732:0.537857:0.163732:0.521423:0.131077:0.521423:0.000000:0.000000:0.000000:0.000000:0.000000 e = :@0.195889:0.537857:0.238226:0.537857:0.238226:0.521423:0.195889:0.521423:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . :@0.270254:0.537857:0.285551:0.537857:0.285551:0.521423:0.270254:0.521423:0.000000:0.000000:0.000000:0.000000 Dx cot x :@0.180409:0.572361:0.238369:0.572361:0.238369:0.555955:0.180409:0.555955:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = – :@0.238355:0.572361:0.266371:0.572361:0.266371:0.555927:0.238355:0.555927:0.000000:0.000000:0.000000:0.000000 csc x Dxx:@0.266260:0.572361:0.341806:0.572379:0.341806:0.555973:0.266260:0.555955:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.287756:0.566316:0.292993:0.566316:0.292993:0.556735:0.287756:0.556735:0.000000 :@0.292946:0.572379:0.312273:0.572379:0.312273:0.555946:0.292946:0.555946:0.000000:0.000000:0.000000:0.000000 = :@0.341762:0.572379:0.360647:0.572379:0.360647:0.555946:0.341762:0.555946:0.000000:0.000000:0.000000 sec x tan x Dx csc x:@0.360555:0.572379:0.496077:0.572379:0.496077:0.555973:0.360555:0.555973:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.430613:0.572379:0.438602:0.572379:0.438602:0.555946:0.430613:0.555946:0.000000:0.000000 = – :@0.496070:0.572379:0.528043:0.572379:0.528043:0.555946:0.496070:0.555946:0.000000:0.000000:0.000000:0.000000:0.000000 csc x cot x:@0.527933:0.572379:0.595807:0.572379:0.595807:0.555973:0.527933:0.555973:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . :@0.595800:0.572379:0.603034:0.572379:0.603034:0.555946:0.595800:0.555946:0.000000:0.000000 Ejemplo 3. :@0.073089:0.606884:0.156437:0.606884:0.156437:0.590214:0.073089:0.590214:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Determina la derivada de la función = :@0.156309:0.606884:0.445063:0.606884:0.445063:0.590450:0.156309:0.590450:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.418778:0.606884:0.426251:0.606884:0.426251:0.590477:0.418778:0.590477:0.000000 sen x tan x:@0.444971:0.606884:0.519060:0.606884:0.519060:0.590477:0.444971:0.590477:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.468422:0.606884:0.483792:0.606884:0.483792:0.590450:0.468422:0.590450:0.000000:0.000000:0.000000 . :@0.507648:0.606884:0.526221:0.606884:0.526221:0.590450:0.507648:0.590450:0.000000:0.000000:0.000000:0.000000 Solución: :@0.073071:0.634246:0.143453:0.634246:0.143453:0.617812:0.073071:0.617812:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 solo debemos aplicar la regla para derivar un producto de funciones.:@0.143313:0.634246:0.644739:0.634246:0.644739:0.617812:0.143313:0.617812:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.073071:0.658723:0.116678:0.658723:0.116678:0.642289:0.073071:0.642289:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.122535:0.651001:0.139193:0.651001:0.139193:0.634595:0.122535:0.634595:0.000000:0.000000 d:@0.122167:0.667393:0.132180:0.667393:0.132180:0.650959:0.122167:0.650959:0.000000 x:@0.132107:0.667393:0.139562:0.667393:0.139562:0.650987:0.132107:0.650987:0.000000 = :@0.145102:0.658739:0.163988:0.658739:0.163988:0.642305:0.145102:0.642305:0.000000:0.000000:0.000000 sen x:@0.163896:0.658739:0.198759:0.658739:0.198759:0.642333:0.163896:0.642333:0.000000:0.000000:0.000000:0.000000:0.000000 sec + :@0.187347:0.658739:0.257164:0.658733:0.257164:0.642299:0.187347:0.642305:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.226407:0.652669:0.231644:0.652669:0.231644:0.643088:0.226407:0.643088:0.000000 x tan x cos x:@0.231596:0.658733:0.330572:0.658733:0.330572:0.642327:0.231596:0.642327:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.281001:0.658733:0.296371:0.658733:0.296371:0.642299:0.281001:0.642299:0.000000:0.000000:0.000000 :@0.319159:0.658733:0.334603:0.658733:0.334603:0.642299:0.319159:0.642299:0.000000:0.000000:0.000000 = :@0.073091:0.678953:0.163213:0.678953:0.163213:0.662519:0.073091:0.662519:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 sen x sec x + senx:@0.162992:0.678953:0.282776:0.678949:0.282776:0.662543:0.162992:0.662547:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.221942:0.672885:0.227179:0.672885:0.227179:0.663304:0.221942:0.663304:0.000000 :@0.282832:0.678949:0.286863:0.678949:0.286863:0.662515:0.282832:0.662515:0.000000 = :@0.073099:0.699169:0.163221:0.699169:0.163221:0.682735:0.073099:0.682735:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 sen x :@0.163000:0.699169:0.201103:0.699169:0.201103:0.682763:0.163000:0.682763:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (sec + 1).:@0.186451:0.699169:0.277189:0.699164:0.277189:0.682730:0.186451:0.682735:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.229589:0.693100:0.234826:0.693100:0.234826:0.683519:0.229589:0.683519:0.000000 x :@0.234779:0.699164:0.245473:0.699164:0.245473:0.682758:0.234779:0.682758:0.000000:0.000000 Ejemplo 4.:@0.073089:0.738175:0.152609:0.738175:0.152609:0.721505:0.073089:0.721505:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Determina :@0.152498:0.738175:0.237959:0.738175:0.237959:0.721741:0.152498:0.721741:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dy:@0.243813:0.732233:0.260472:0.732233:0.260472:0.715827:0.243813:0.715827:0.000000:0.000000 d:@0.243445:0.748625:0.253459:0.748625:0.253459:0.732192:0.243445:0.732192:0.000000 x:@0.253385:0.748625:0.260840:0.748625:0.260840:0.732219:0.253385:0.732219:0.000000 , si = :@0.266381:0.738183:0.318197:0.738183:0.318197:0.721749:0.266381:0.721749:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.291911:0.738183:0.299385:0.738183:0.299385:0.721777:0.291911:0.721777:0.000000 csox:@0.325118:0.730208:0.355424:0.730208:0.355424:0.713802:0.325118:0.713802:0.000000:0.000000:0.000000:0.000000 1–2:@0.323093:0.746850:0.350115:0.746850:0.350115:0.730416:0.323093:0.730416:0.000000:0.000000:0.000000 x:@0.350023:0.746850:0.357478:0.746850:0.357478:0.730444:0.350023:0.730444:0.000000 .:@0.362374:0.738183:0.365577:0.738183:0.365577:0.721749:0.362374:0.721749:0.000000 Solución::@0.073104:0.765545:0.139529:0.765545:0.139529:0.749111:0.073104:0.749111:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 solo debemos aplicar la regla para derivar un producto de funciones.:@0.139407:0.765545:0.644791:0.765545:0.644791:0.749111:0.139407:0.749111:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.883029:0.306645:0.888805:0.306645:0.888805:0.296448:0.883029:0.296448:0.000000 x:@0.908879:0.313328:0.915086:0.313328:0.915086:0.300571:0.908879:0.300571:0.000000 5:@0.715661:0.229250:0.721438:0.229250:0.721438:0.219054:0.715661:0.219054:0.000000 y:@0.698648:0.192460:0.704855:0.192460:0.704855:0.179704:0.698648:0.179704:0.000000 ( , 0):@0.839015:0.293251:0.864034:0.293251:0.864034:0.283054:0.839015:0.283054:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.842213:0.293607:0.850342:0.293607:0.850342:0.282808:0.842213:0.282808:0.000000 y = Sen2x:@0.734301:0.310437:0.779587:0.310437:0.779587:0.300241:0.734301:0.300241:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Recta tangente:@0.841976:0.187637:0.913293:0.187637:0.913293:0.177440:0.841976:0.177440:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Notación de la derivada de una función :@0.177840:0.253579:0.468262:0.253579:0.468262:0.237145:0.177840:0.237145:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y=f(x):@0.467923:0.253579:0.507904:0.253579:0.507904:0.237173:0.467923:0.237173:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.264413:0.273799:0.271886:0.273799:0.271886:0.257392:0.264413:0.257392:0.000000 ´= ´( ) = :@0.271812:0.273799:0.338023:0.273799:0.338023:0.257365:0.271812:0.257365:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.292097:0.273799:0.314407:0.273799:0.314407:0.257392:0.292097:0.257392:0.000000:0.000000:0.000000 df/dx= dy/dx:@0.337913:0.273799:0.427538:0.273799:0.427538:0.257392:0.337913:0.257392:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.427611:0.273799:0.431643:0.273799:0.431643:0.257365:0.427611:0.257365:0.000000 Todas indican la derivada de respecto a .:@0.177788:0.294018:0.487932:0.294018:0.487932:0.277585:0.177788:0.277585:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.386582:0.294018:0.394055:0.294018:0.394055:0.277612:0.386582:0.277612:0.000000 x:@0.477348:0.294018:0.484803:0.294018:0.484803:0.277612:0.477348:0.277612:0.000000 Competencia :@0.740294:0.418726:0.838985:0.418726:0.838985:0.403320:0.740294:0.403320:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemática :@0.740294:0.433855:0.828908:0.433855:0.828908:0.418449:0.740294:0.418449:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 En el diseño de rampas, :@0.743498:0.462480:0.902865:0.462480:0.902865:0.447541:0.743498:0.447541:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 el análisis de la función :@0.743498:0.477609:0.898504:0.477609:0.898504:0.462669:0.743498:0.462669:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 tangente y su derivada :@0.743498:0.492738:0.897972:0.492738:0.897972:0.477798:0.743498:0.477798:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 es fundamental, no solo :@0.743498:0.507867:0.905275:0.507867:0.905275:0.492927:0.743498:0.492927:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 en la ingeniería y en la :@0.743498:0.522996:0.894180:0.522996:0.894180:0.508056:0.743498:0.508056:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 arquitectura, sino también :@0.743498:0.538125:0.920840:0.538125:0.920840:0.523185:0.743498:0.523185:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 en la creación de entornos :@0.743498:0.553254:0.922562:0.553254:0.922562:0.538314:0.743498:0.538314:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 inclusivos. Una rampa :@0.743498:0.568383:0.890676:0.568383:0.890676:0.553443:0.743498:0.553443:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 bien diseñada reduce :@0.743498:0.583512:0.889697:0.583512:0.889697:0.568572:0.743498:0.568572:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 el riesgo de lesiones y :@0.743498:0.598641:0.891134:0.598641:0.891134:0.583701:0.743498:0.583701:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 los problemas de salud :@0.743498:0.613770:0.899368:0.613770:0.899368:0.598830:0.743498:0.598830:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 derivados de la fatiga o el :@0.743498:0.628899:0.915198:0.628899:0.915198:0.613959:0.743498:0.613959:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 esfuerzo e cesivo. Así, un :@0.743498:0.644028:0.912090:0.644028:0.912090:0.629088:0.743498:0.629088:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.810902:0.644028:0.817679:0.644028:0.817679:0.629113:0.810902:0.629113:0.000000 diseño accesible no solo :@0.743482:0.659157:0.908176:0.659157:0.908176:0.644217:0.743482:0.644217:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 facilita la movilidad, sino :@0.743482:0.674286:0.907060:0.674286:0.907060:0.659346:0.743482:0.659346:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que también protege la :@0.743482:0.689414:0.903519:0.689414:0.903519:0.674475:0.743482:0.674475:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 salud física de los usuarios.:@0.743482:0.704543:0.918253:0.704543:0.918253:0.689604:0.743482:0.689604:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Aplicar la función :@0.743482:0.719672:0.861623:0.719672:0.861623:0.704733:0.743482:0.704733:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 tangente permite calcular :@0.743482:0.734801:0.917795:0.734801:0.917795:0.719861:0.743482:0.719861:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 el ángulo óptimo para :@0.743482:0.749930:0.894254:0.749930:0.894254:0.734990:0.743482:0.734990:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 una rampa. Un ángulo :@0.743482:0.765059:0.894756:0.765059:0.894756:0.750119:0.743482:0.750119:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 demasiado inclinado :@0.743482:0.780188:0.885017:0.780188:0.885017:0.765248:0.743482:0.765248:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dificulta el acceso y :@0.743482:0.795317:0.874759:0.795317:0.874759:0.780377:0.743482:0.780377:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 aumenta el riesgo de :@0.743482:0.810446:0.886700:0.810446:0.886700:0.795506:0.743482:0.795506:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 caídas, mientras que uno :@0.743482:0.825575:0.911884:0.825575:0.911884:0.810635:0.743482:0.810635:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 muy bajo puede requerir :@0.743482:0.840704:0.911811:0.840704:0.911811:0.825764:0.743482:0.825764:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 demasiado espacio, lo cual :@0.743482:0.855833:0.923935:0.855833:0.923935:0.840893:0.743482:0.840893:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 puede no ser práctico en :@0.743482:0.870962:0.912422:0.870962:0.912422:0.856022:0.743482:0.856022:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 todos los entornos. :@0.743482:0.886091:0.873110:0.886091:0.873110:0.871151:0.743482:0.871151:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 p:@0.862006:0.375054:0.873474:0.375054:0.873474:0.365002:0.862006:0.365002:0.000000 :@0.873239:0.375713:0.876171:0.375713:0.876171:0.363761:0.873239:0.363761:0.000000 Figura 3.2:@0.876118:0.375713:0.926956:0.375713:0.926956:0.363761:0.876118:0.363761:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 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