[{"data":1,"prerenderedAt":2845},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fcalculus\u002Ffaktorregel":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fcalculus\u002Ffaktorregel":2838},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":554,"category":101,"checks":2788,"conditions":2803,"conditionsHtml":2804,"deprecated":2805,"description":2782,"extension":2806,"figure":2805,"figureCaption":2805,"figureHtml":2805,"formulaId":2807,"latex":2808,"latexHtml":2809,"level":2810,"meta":2811,"navigation":2812,"path":2813,"redirectFrom":2814,"related":2815,"seo":2818,"stem":2819,"subject":584,"summary":2820,"tags":2821,"variables":2826,"__hash__":2837},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fcalculus\u002Ffaktorregel.md","Faktorregel",[550,551,552,553],"konstanter Faktor Ableitung","Faktorregel Ableitung","Zahl vor x ableiten","Ableitung mit Vorfaktor",{"type":555,"value":556,"toc":2781},"minimark",[557,562,571,711,989,993,1340,1636,2057,2061],[558,559,561],"h2",{"id":560},"erklärung","Erklärung",[563,564,565,566,570],"p",{},"Steht vor einer Funktion eine feste Zahl als ",[567,568,569],"strong",{},"Faktor",", wird nur die Funktion abgeleitet. Der Faktor wird unverändert mitgeschrieben.",[563,572,573,574,620,621,652,653,681,682,710],{},"Anschaulich: Multipliziert man alle Funktionswerte mit ",[575,576,579,601],"span",{"className":577},[578],"katex",[575,580,583],{"className":581},[582],"katex-mathml",[584,585,587],"math",{"xmlns":586},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[588,589,590,597],"semantics",{},[591,592,593],"mrow",{},[594,595,596],"mi",{},"c",[598,599,596],"annotation",{"encoding":600},"application\u002Fx-tex",[575,602,606],{"className":603,"ariaHidden":605},[604],"katex-html","true",[575,607,610,615],{"className":608},[609],"base",[575,611],{"className":612,"style":614},[613],"strut","height:0.4306em;",[575,616,596],{"className":617},[618,619],"mord","mathnormal",", wird der Graph in ",[575,622,624,638],{"className":623},[578],[575,625,627],{"className":626},[582],[584,628,629],{"xmlns":586},[588,630,631,636],{},[591,632,633],{},[594,634,635],{},"y",[598,637,635],{"encoding":600},[575,639,641],{"className":640,"ariaHidden":605},[604],[575,642,644,648],{"className":643},[609],[575,645],{"className":646,"style":647},[613],"height:0.625em;vertical-align:-0.1944em;",[575,649,635],{"className":650,"style":651},[618,619],"margin-right:0.0359em;","-Richtung um den Faktor ",[575,654,656,669],{"className":655},[578],[575,657,659],{"className":658},[582],[584,660,661],{"xmlns":586},[588,662,663,667],{},[591,664,665],{},[594,666,596],{},[598,668,596],{"encoding":600},[575,670,672],{"className":671,"ariaHidden":605},[604],[575,673,675,678],{"className":674},[609],[575,676],{"className":677,"style":614},[613],[575,679,596],{"className":680},[618,619]," gestreckt. Alle Steigungen werden dabei ebenfalls mit ",[575,683,685,698],{"className":684},[578],[575,686,688],{"className":687},[582],[584,689,690],{"xmlns":586},[588,691,692,696],{},[591,693,694],{},[594,695,596],{},[598,697,596],{"encoding":600},[575,699,701],{"className":700,"ariaHidden":605},[604],[575,702,704,707],{"className":703},[609],[575,705],{"className":706,"style":614},[613],[575,708,596],{"className":709},[618,619]," multipliziert.",[563,712,713,714,717,718,799,800,836,837,924,925,953,954,988],{},"Wichtig ist der Unterschied zu einem konstanten ",[567,715,716],{},"Summanden",": ",[575,719,721,746],{"className":720},[578],[575,722,724],{"className":723},[582],[584,725,726],{"xmlns":586},[588,727,728,743],{},[591,729,730,734],{},[731,732,733],"mn",{},"5",[735,736,737,740],"msup",{},[594,738,739],{},"x",[731,741,742],{},"2",[598,744,745],{"encoding":600},"5x^2",[575,747,749],{"className":748,"ariaHidden":605},[604],[575,750,752,756,759],{"className":751},[609],[575,753],{"className":754,"style":755},[613],"height:0.8141em;",[575,757,733],{"className":758},[618],[575,760,762,765],{"className":761},[618],[575,763,739],{"className":764},[618,619],[575,766,769],{"className":767},[768],"msupsub",[575,770,773],{"className":771},[772],"vlist-t",[575,774,777],{"className":775},[776],"vlist-r",[575,778,781],{"className":779,"style":755},[780],"vlist",[575,782,784,789],{"style":783},"top:-3.063em;margin-right:0.05em;",[575,785],{"className":786,"style":788},[787],"pstrut","height:2.7em;",[575,790,796],{"className":791},[792,793,794,795],"sizing","reset-size6","size3","mtight",[575,797,742],{"className":798},[618,795]," hat die Ableitung ",[575,801,803,820],{"className":802},[578],[575,804,806],{"className":805},[582],[584,807,808],{"xmlns":586},[588,809,810,817],{},[591,811,812,815],{},[731,813,814],{},"10",[594,816,739],{},[598,818,819],{"encoding":600},"10x",[575,821,823],{"className":822,"ariaHidden":605},[604],[575,824,826,830,833],{"className":825},[609],[575,827],{"className":828,"style":829},[613],"height:0.6444em;",[575,831,814],{"className":832},[618],[575,834,739],{"className":835},[618,619],", bei ",[575,838,840,864],{"className":839},[578],[575,841,843],{"className":842},[582],[584,844,845],{"xmlns":586},[588,846,847,861],{},[591,848,849,855,859],{},[735,850,851,853],{},[594,852,739],{},[731,854,742],{},[856,857,858],"mo",{},"+",[731,860,733],{},[598,862,863],{"encoding":600},"x^2 + 5",[575,865,867,915],{"className":866,"ariaHidden":605},[604],[575,868,870,874,903,908,912],{"className":869},[609],[575,871],{"className":872,"style":873},[613],"height:0.8974em;vertical-align:-0.0833em;",[575,875,877,880],{"className":876},[618],[575,878,739],{"className":879},[618,619],[575,881,883],{"className":882},[768],[575,884,886],{"className":885},[772],[575,887,889],{"className":888},[776],[575,890,892],{"className":891,"style":755},[780],[575,893,894,897],{"style":783},[575,895],{"className":896,"style":788},[787],[575,898,900],{"className":899},[792,793,794,795],[575,901,742],{"className":902},[618,795],[575,904],{"className":905,"style":907},[906],"mspace","margin-right:0.2222em;",[575,909,858],{"className":910},[911],"mbin",[575,913],{"className":914,"style":907},[906],[575,916,918,921],{"className":917},[609],[575,919],{"className":920,"style":829},[613],[575,922,733],{"className":923},[618]," fällt die ",[575,926,928,941],{"className":927},[578],[575,929,931],{"className":930},[582],[584,932,933],{"xmlns":586},[588,934,935,939],{},[591,936,937],{},[731,938,733],{},[598,940,733],{"encoding":600},[575,942,944],{"className":943,"ariaHidden":605},[604],[575,945,947,950],{"className":946},[609],[575,948],{"className":949,"style":829},[613],[575,951,733],{"className":952},[618]," dagegen weg, die Ableitung ist ",[575,955,957,973],{"className":956},[578],[575,958,960],{"className":959},[582],[584,961,962],{"xmlns":586},[588,963,964,970],{},[591,965,966,968],{},[731,967,742],{},[594,969,739],{},[598,971,972],{"encoding":600},"2x",[575,974,976],{"className":975,"ariaHidden":605},[604],[575,977,979,982,985],{"className":978},[609],[575,980],{"className":981,"style":829},[613],[575,983,742],{"className":984},[618],[575,986,739],{"className":987},[618,619],".",[558,990,992],{"id":991},"beispiele","Beispiele",[575,994,997],{"className":995},[996],"katex-display",[575,998,1000,1088],{"className":999},[578],[575,1001,1003],{"className":1002},[582],[584,1004,1006],{"xmlns":586,"display":1005},"block",[588,1007,1008,1085],{},[591,1009,1010,1013,1017,1019,1022,1025,1028,1035,1039,1042,1044,1053,1055,1057,1059,1061,1063,1066,1068,1074,1076,1079],{},[594,1011,1012],{},"f",[856,1014,1016],{"stretchy":1015},"false","(",[594,1018,739],{},[856,1020,1021],{"stretchy":1015},")",[856,1023,1024],{},"=",[731,1026,1027],{},"4",[735,1029,1030,1032],{},[594,1031,739],{},[731,1033,1034],{},"3",[1036,1037,1038],"mtext",{},"  ",[856,1040,1041],{},"⇒",[1036,1043,1038],{},[735,1045,1046,1048],{},[594,1047,1012],{},[856,1049,1052],{"mathvariant":1050,"lspace":1051,"rspace":1051},"normal","0em","′",[856,1054,1016],{"stretchy":1015},[594,1056,739],{},[856,1058,1021],{"stretchy":1015},[856,1060,1024],{},[731,1062,1027],{},[856,1064,1065],{},"⋅",[731,1067,1034],{},[735,1069,1070,1072],{},[594,1071,739],{},[731,1073,742],{},[856,1075,1024],{},[731,1077,1078],{},"12",[735,1080,1081,1083],{},[594,1082,739],{},[731,1084,742],{},[598,1086,1087],{"encoding":600},"f(x) = 4x^3 \\;\\Rightarrow\\; f'(x) = 4 \\cdot 3x^2 = 12x^2",[575,1089,1091,1124,1179,1237,1255,1302],{"className":1090,"ariaHidden":605},[604],[575,1092,1094,1098,1102,1106,1109,1113,1117,1121],{"className":1093},[609],[575,1095],{"className":1096,"style":1097},[613],"height:1em;vertical-align:-0.25em;",[575,1099,1012],{"className":1100,"style":1101},[618,619],"margin-right:0.1076em;",[575,1103,1016],{"className":1104},[1105],"mopen",[575,1107,739],{"className":1108},[618,619],[575,1110,1021],{"className":1111},[1112],"mclose",[575,1114],{"className":1115,"style":1116},[906],"margin-right:0.2778em;",[575,1118,1024],{"className":1119},[1120],"mrel",[575,1122],{"className":1123,"style":1116},[906],[575,1125,1127,1131,1134,1164,1167,1170,1173,1176],{"className":1126},[609],[575,1128],{"className":1129,"style":1130},[613],"height:0.8641em;",[575,1132,1027],{"className":1133},[618],[575,1135,1137,1140],{"className":1136},[618],[575,1138,739],{"className":1139},[618,619],[575,1141,1143],{"className":1142},[768],[575,1144,1146],{"className":1145},[772],[575,1147,1149],{"className":1148},[776],[575,1150,1152],{"className":1151,"style":1130},[780],[575,1153,1155,1158],{"style":1154},"top:-3.113em;margin-right:0.05em;",[575,1156],{"className":1157,"style":788},[787],[575,1159,1161],{"className":1160},[792,793,794,795],[575,1162,1034],{"className":1163},[618,795],[575,1165],{"className":1166,"style":1116},[906],[575,1168],{"className":1169,"style":1116},[906],[575,1171,1041],{"className":1172},[1120],[575,1174],{"className":1175,"style":1116},[906],[575,1177],{"className":1178,"style":1116},[906],[575,1180,1182,1186,1219,1222,1225,1228,1231,1234],{"className":1181},[609],[575,1183],{"className":1184,"style":1185},[613],"height:1.0519em;vertical-align:-0.25em;",[575,1187,1189,1192],{"className":1188},[618],[575,1190,1012],{"className":1191,"style":1101},[618,619],[575,1193,1195],{"className":1194},[768],[575,1196,1198],{"className":1197},[772],[575,1199,1201],{"className":1200},[776],[575,1202,1205],{"className":1203,"style":1204},[780],"height:0.8019em;",[575,1206,1207,1210],{"style":1154},[575,1208],{"className":1209,"style":788},[787],[575,1211,1213],{"className":1212},[792,793,794,795],[575,1214,1216],{"className":1215},[618,795],[575,1217,1052],{"className":1218},[618,795],[575,1220,1016],{"className":1221},[1105],[575,1223,739],{"className":1224},[618,619],[575,1226,1021],{"className":1227},[1112],[575,1229],{"className":1230,"style":1116},[906],[575,1232,1024],{"className":1233},[1120],[575,1235],{"className":1236,"style":1116},[906],[575,1238,1240,1243,1246,1249,1252],{"className":1239},[609],[575,1241],{"className":1242,"style":829},[613],[575,1244,1027],{"className":1245},[618],[575,1247],{"className":1248,"style":907},[906],[575,1250,1065],{"className":1251},[911],[575,1253],{"className":1254,"style":907},[906],[575,1256,1258,1261,1264,1293,1296,1299],{"className":1257},[609],[575,1259],{"className":1260,"style":1130},[613],[575,1262,1034],{"className":1263},[618],[575,1265,1267,1270],{"className":1266},[618],[575,1268,739],{"className":1269},[618,619],[575,1271,1273],{"className":1272},[768],[575,1274,1276],{"className":1275},[772],[575,1277,1279],{"className":1278},[776],[575,1280,1282],{"className":1281,"style":1130},[780],[575,1283,1284,1287],{"style":1154},[575,1285],{"className":1286,"style":788},[787],[575,1288,1290],{"className":1289},[792,793,794,795],[575,1291,742],{"className":1292},[618,795],[575,1294],{"className":1295,"style":1116},[906],[575,1297,1024],{"className":1298},[1120],[575,1300],{"className":1301,"style":1116},[906],[575,1303,1305,1308,1311],{"className":1304},[609],[575,1306],{"className":1307,"style":1130},[613],[575,1309,1078],{"className":1310},[618],[575,1312,1314,1317],{"className":1313},[618],[575,1315,739],{"className":1316},[618,619],[575,1318,1320],{"className":1319},[768],[575,1321,1323],{"className":1322},[772],[575,1324,1326],{"className":1325},[776],[575,1327,1329],{"className":1328,"style":1130},[780],[575,1330,1331,1334],{"style":1154},[575,1332],{"className":1333,"style":788},[787],[575,1335,1337],{"className":1336},[792,793,794,795],[575,1338,742],{"className":1339},[618,795],[575,1341,1343],{"className":1342},[996],[575,1344,1346,1416],{"className":1345},[578],[575,1347,1349],{"className":1348},[582],[584,1350,1351],{"xmlns":586,"display":1005},[588,1352,1353,1413],{},[591,1354,1355,1357,1359,1361,1363,1365,1368,1371,1377,1379,1381,1383,1389,1391,1393,1395,1397,1399,1401,1403,1405,1407,1409,1411],{},[594,1356,1012],{},[856,1358,1016],{"stretchy":1015},[594,1360,739],{},[856,1362,1021],{"stretchy":1015},[856,1364,1024],{},[856,1366,1367],{},"−",[731,1369,1370],{},"0,5",[735,1372,1373,1375],{},[594,1374,739],{},[731,1376,742],{},[1036,1378,1038],{},[856,1380,1041],{},[1036,1382,1038],{},[735,1384,1385,1387],{},[594,1386,1012],{},[856,1388,1052],{"mathvariant":1050,"lspace":1051,"rspace":1051},[856,1390,1016],{"stretchy":1015},[594,1392,739],{},[856,1394,1021],{"stretchy":1015},[856,1396,1024],{},[856,1398,1367],{},[731,1400,1370],{},[856,1402,1065],{},[731,1404,742],{},[594,1406,739],{},[856,1408,1024],{},[856,1410,1367],{},[594,1412,739],{},[598,1414,1415],{"encoding":600},"f(x) = -0{,}5x^2 \\;\\Rightarrow\\; f'(x) = -0{,}5 \\cdot 2x = -x",[575,1417,1419,1446,1515,1571,1602,1623],{"className":1418,"ariaHidden":605},[604],[575,1420,1422,1425,1428,1431,1434,1437,1440,1443],{"className":1421},[609],[575,1423],{"className":1424,"style":1097},[613],[575,1426,1012],{"className":1427,"style":1101},[618,619],[575,1429,1016],{"className":1430},[1105],[575,1432,739],{"className":1433},[618,619],[575,1435,1021],{"className":1436},[1112],[575,1438],{"className":1439,"style":1116},[906],[575,1441,1024],{"className":1442},[1120],[575,1444],{"className":1445,"style":1116},[906],[575,1447,1449,1453,1456,1460,1468,1471,1500,1503,1506,1509,1512],{"className":1448},[609],[575,1450],{"className":1451,"style":1452},[613],"height:1.0585em;vertical-align:-0.1944em;",[575,1454,1367],{"className":1455},[618],[575,1457,1459],{"className":1458},[618],"0",[575,1461,1463],{"className":1462},[618],[575,1464,1467],{"className":1465},[1466],"mpunct",",",[575,1469,733],{"className":1470},[618],[575,1472,1474,1477],{"className":1473},[618],[575,1475,739],{"className":1476},[618,619],[575,1478,1480],{"className":1479},[768],[575,1481,1483],{"className":1482},[772],[575,1484,1486],{"className":1485},[776],[575,1487,1489],{"className":1488,"style":1130},[780],[575,1490,1491,1494],{"style":1154},[575,1492],{"className":1493,"style":788},[787],[575,1495,1497],{"className":1496},[792,793,794,795],[575,1498,742],{"className":1499},[618,795],[575,1501],{"className":1502,"style":1116},[906],[575,1504],{"className":1505,"style":1116},[906],[575,1507,1041],{"className":1508},[1120],[575,1510],{"className":1511,"style":1116},[906],[575,1513],{"className":1514,"style":1116},[906],[575,1516,1518,1521,1553,1556,1559,1562,1565,1568],{"className":1517},[609],[575,1519],{"className":1520,"style":1185},[613],[575,1522,1524,1527],{"className":1523},[618],[575,1525,1012],{"className":1526,"style":1101},[618,619],[575,1528,1530],{"className":1529},[768],[575,1531,1533],{"className":1532},[772],[575,1534,1536],{"className":1535},[776],[575,1537,1539],{"className":1538,"style":1204},[780],[575,1540,1541,1544],{"style":1154},[575,1542],{"className":1543,"style":788},[787],[575,1545,1547],{"className":1546},[792,793,794,795],[575,1548,1550],{"className":1549},[618,795],[575,1551,1052],{"className":1552},[618,795],[575,1554,1016],{"className":1555},[1105],[575,1557,739],{"className":1558},[618,619],[575,1560,1021],{"className":1561},[1112],[575,1563],{"className":1564,"style":1116},[906],[575,1566,1024],{"className":1567},[1120],[575,1569],{"className":1570,"style":1116},[906],[575,1572,1574,1578,1581,1584,1590,1593,1596,1599],{"className":1573},[609],[575,1575],{"className":1576,"style":1577},[613],"height:0.8389em;vertical-align:-0.1944em;",[575,1579,1367],{"className":1580},[618],[575,1582,1459],{"className":1583},[618],[575,1585,1587],{"className":1586},[618],[575,1588,1467],{"className":1589},[1466],[575,1591,733],{"className":1592},[618],[575,1594],{"className":1595,"style":907},[906],[575,1597,1065],{"className":1598},[911],[575,1600],{"className":1601,"style":907},[906],[575,1603,1605,1608,1611,1614,1617,1620],{"className":1604},[609],[575,1606],{"className":1607,"style":829},[613],[575,1609,742],{"className":1610},[618],[575,1612,739],{"className":1613},[618,619],[575,1615],{"className":1616,"style":1116},[906],[575,1618,1024],{"className":1619},[1120],[575,1621],{"className":1622,"style":1116},[906],[575,1624,1626,1630,1633],{"className":1625},[609],[575,1627],{"className":1628,"style":1629},[613],"height:0.6667em;vertical-align:-0.0833em;",[575,1631,1367],{"className":1632},[618],[575,1634,739],{"className":1635},[618,619],[575,1637,1639],{"className":1638},[996],[575,1640,1642,1710],{"className":1641},[578],[575,1643,1645],{"className":1644},[582],[584,1646,1647],{"xmlns":586,"display":1005},[588,1648,1649,1707],{},[591,1650,1651,1653,1655,1657,1659,1661,1668,1670,1677,1679,1681,1683,1685,1687,1693,1695,1697,1699,1701],{},[594,1652,1012],{},[856,1654,1016],{"stretchy":1015},[594,1656,739],{},[856,1658,1021],{"stretchy":1015},[856,1660,1024],{},[1662,1663,1664,1666],"mfrac",{},[594,1665,739],{},[731,1667,1034],{},[856,1669,1024],{},[1662,1671,1672,1675],{},[731,1673,1674],{},"1",[731,1676,1034],{},[856,1678,1065],{},[594,1680,739],{},[1036,1682,1038],{},[856,1684,1041],{},[1036,1686,1038],{},[735,1688,1689,1691],{},[594,1690,1012],{},[856,1692,1052],{"mathvariant":1050,"lspace":1051,"rspace":1051},[856,1694,1016],{"stretchy":1015},[594,1696,739],{},[856,1698,1021],{"stretchy":1015},[856,1700,1024],{},[1662,1702,1703,1705],{},[731,1704,1674],{},[731,1706,1034],{},[598,1708,1709],{"encoding":600},"f(x) = \\frac{x}{3} = \\frac{1}{3} \\cdot x \\;\\Rightarrow\\; f'(x) = \\frac{1}{3}",[575,1711,1713,1740,1830,1909,1933,1989],{"className":1712,"ariaHidden":605},[604],[575,1714,1716,1719,1722,1725,1728,1731,1734,1737],{"className":1715},[609],[575,1717],{"className":1718,"style":1097},[613],[575,1720,1012],{"className":1721,"style":1101},[618,619],[575,1723,1016],{"className":1724},[1105],[575,1726,739],{"className":1727},[618,619],[575,1729,1021],{"className":1730},[1112],[575,1732],{"className":1733,"style":1116},[906],[575,1735,1024],{"className":1736},[1120],[575,1738],{"className":1739,"style":1116},[906],[575,1741,1743,1747,1821,1824,1827],{"className":1742},[609],[575,1744],{"className":1745,"style":1746},[613],"height:1.7936em;vertical-align:-0.686em;",[575,1748,1750,1754,1818],{"className":1749},[618],[575,1751],{"className":1752},[1105,1753],"nulldelimiter",[575,1755,1757],{"className":1756},[1662],[575,1758,1761,1809],{"className":1759},[772,1760],"vlist-t2",[575,1762,1764,1804],{"className":1763},[776],[575,1765,1768,1781,1792],{"className":1766,"style":1767},[780],"height:1.1076em;",[575,1769,1771,1775],{"style":1770},"top:-2.314em;",[575,1772],{"className":1773,"style":1774},[787],"height:3em;",[575,1776,1778],{"className":1777},[618],[575,1779,1034],{"className":1780},[618],[575,1782,1784,1787],{"style":1783},"top:-3.23em;",[575,1785],{"className":1786,"style":1774},[787],[575,1788],{"className":1789,"style":1791},[1790],"frac-line","border-bottom-width:0.04em;",[575,1793,1795,1798],{"style":1794},"top:-3.677em;",[575,1796],{"className":1797,"style":1774},[787],[575,1799,1801],{"className":1800},[618],[575,1802,739],{"className":1803},[618,619],[575,1805,1808],{"className":1806},[1807],"vlist-s","​",[575,1810,1812],{"className":1811},[776],[575,1813,1816],{"className":1814,"style":1815},[780],"height:0.686em;",[575,1817],{},[575,1819],{"className":1820},[1112,1753],[575,1822],{"className":1823,"style":1116},[906],[575,1825,1024],{"className":1826},[1120],[575,1828],{"className":1829,"style":1116},[906],[575,1831,1833,1837,1900,1903,1906],{"className":1832},[609],[575,1834],{"className":1835,"style":1836},[613],"height:2.0074em;vertical-align:-0.686em;",[575,1838,1840,1843,1897],{"className":1839},[618],[575,1841],{"className":1842},[1105,1753],[575,1844,1846],{"className":1845},[1662],[575,1847,1849,1889],{"className":1848},[772,1760],[575,1850,1852,1886],{"className":1851},[776],[575,1853,1856,1867,1875],{"className":1854,"style":1855},[780],"height:1.3214em;",[575,1857,1858,1861],{"style":1770},[575,1859],{"className":1860,"style":1774},[787],[575,1862,1864],{"className":1863},[618],[575,1865,1034],{"className":1866},[618],[575,1868,1869,1872],{"style":1783},[575,1870],{"className":1871,"style":1774},[787],[575,1873],{"className":1874,"style":1791},[1790],[575,1876,1877,1880],{"style":1794},[575,1878],{"className":1879,"style":1774},[787],[575,1881,1883],{"className":1882},[618],[575,1884,1674],{"className":1885},[618],[575,1887,1808],{"className":1888},[1807],[575,1890,1892],{"className":1891},[776],[575,1893,1895],{"className":1894,"style":1815},[780],[575,1896],{},[575,1898],{"className":1899},[1112,1753],[575,1901],{"className":1902,"style":907},[906],[575,1904,1065],{"className":1905},[911],[575,1907],{"className":1908,"style":907},[906],[575,1910,1912,1915,1918,1921,1924,1927,1930],{"className":1911},[609],[575,1913],{"className":1914,"style":614},[613],[575,1916,739],{"className":1917},[618,619],[575,1919],{"className":1920,"style":1116},[906],[575,1922],{"className":1923,"style":1116},[906],[575,1925,1041],{"className":1926},[1120],[575,1928],{"className":1929,"style":1116},[906],[575,1931],{"className":1932,"style":1116},[906],[575,1934,1936,1939,1971,1974,1977,1980,1983,1986],{"className":1935},[609],[575,1937],{"className":1938,"style":1185},[613],[575,1940,1942,1945],{"className":1941},[618],[575,1943,1012],{"className":1944,"style":1101},[618,619],[575,1946,1948],{"className":1947},[768],[575,1949,1951],{"className":1950},[772],[575,1952,1954],{"className":1953},[776],[575,1955,1957],{"className":1956,"style":1204},[780],[575,1958,1959,1962],{"style":1154},[575,1960],{"className":1961,"style":788},[787],[575,1963,1965],{"className":1964},[792,793,794,795],[575,1966,1968],{"className":1967},[618,795],[575,1969,1052],{"className":1970},[618,795],[575,1972,1016],{"className":1973},[1105],[575,1975,739],{"className":1976},[618,619],[575,1978,1021],{"className":1979},[1112],[575,1981],{"className":1982,"style":1116},[906],[575,1984,1024],{"className":1985},[1120],[575,1987],{"className":1988,"style":1116},[906],[575,1990,1992,1995],{"className":1991},[609],[575,1993],{"className":1994,"style":1836},[613],[575,1996,1998,2001,2054],{"className":1997},[618],[575,1999],{"className":2000},[1105,1753],[575,2002,2004],{"className":2003},[1662],[575,2005,2007,2046],{"className":2006},[772,1760],[575,2008,2010,2043],{"className":2009},[776],[575,2011,2013,2024,2032],{"className":2012,"style":1855},[780],[575,2014,2015,2018],{"style":1770},[575,2016],{"className":2017,"style":1774},[787],[575,2019,2021],{"className":2020},[618],[575,2022,1034],{"className":2023},[618],[575,2025,2026,2029],{"style":1783},[575,2027],{"className":2028,"style":1774},[787],[575,2030],{"className":2031,"style":1791},[1790],[575,2033,2034,2037],{"style":1794},[575,2035],{"className":2036,"style":1774},[787],[575,2038,2040],{"className":2039},[618],[575,2041,1674],{"className":2042},[618],[575,2044,1808],{"className":2045},[1807],[575,2047,2049],{"className":2048},[776],[575,2050,2052],{"className":2051,"style":1815},[780],[575,2053],{},[575,2055],{"className":2056},[1112,1753],[558,2058,2060],{"id":2059},"typische-fehler","Typische Fehler",[2062,2063,2064,2235,2526],"ul",{},[2065,2066,2067,2068,2234],"li",{},"Den Faktor beim Ableiten weglassen: ",[575,2069,2071,2109],{"className":2070},[578],[575,2072,2074],{"className":2073},[582],[584,2075,2076],{"xmlns":586},[588,2077,2078,2106],{},[591,2079,2080,2082,2084,2090,2096,2098,2100],{},[856,2081,1016],{"stretchy":1015},[731,2083,1027],{},[735,2085,2086,2088],{},[594,2087,739],{},[731,2089,1034],{},[735,2091,2092,2094],{},[856,2093,1021],{"stretchy":1015},[856,2095,1052],{"mathvariant":1050,"lspace":1051,"rspace":1051},[856,2097,1024],{},[731,2099,1034],{},[735,2101,2102,2104],{},[594,2103,739],{},[731,2105,742],{},[598,2107,2108],{"encoding":600},"(4x^3)' = 3x^2",[575,2110,2112,2196],{"className":2111,"ariaHidden":605},[604],[575,2113,2115,2119,2122,2125,2154,2187,2190,2193],{"className":2114},[609],[575,2116],{"className":2117,"style":2118},[613],"height:1.0641em;vertical-align:-0.25em;",[575,2120,1016],{"className":2121},[1105],[575,2123,1027],{"className":2124},[618],[575,2126,2128,2131],{"className":2127},[618],[575,2129,739],{"className":2130},[618,619],[575,2132,2134],{"className":2133},[768],[575,2135,2137],{"className":2136},[772],[575,2138,2140],{"className":2139},[776],[575,2141,2143],{"className":2142,"style":755},[780],[575,2144,2145,2148],{"style":783},[575,2146],{"className":2147,"style":788},[787],[575,2149,2151],{"className":2150},[792,793,794,795],[575,2152,1034],{"className":2153},[618,795],[575,2155,2157,2160],{"className":2156},[1112],[575,2158,1021],{"className":2159},[1112],[575,2161,2163],{"className":2162},[768],[575,2164,2166],{"className":2165},[772],[575,2167,2169],{"className":2168},[776],[575,2170,2173],{"className":2171,"style":2172},[780],"height:0.7519em;",[575,2174,2175,2178],{"style":783},[575,2176],{"className":2177,"style":788},[787],[575,2179,2181],{"className":2180},[792,793,794,795],[575,2182,2184],{"className":2183},[618,795],[575,2185,1052],{"className":2186},[618,795],[575,2188],{"className":2189,"style":1116},[906],[575,2191,1024],{"className":2192},[1120],[575,2194],{"className":2195,"style":1116},[906],[575,2197,2199,2202,2205],{"className":2198},[609],[575,2200],{"className":2201,"style":755},[613],[575,2203,1034],{"className":2204},[618],[575,2206,2208,2211],{"className":2207},[618],[575,2209,739],{"className":2210},[618,619],[575,2212,2214],{"className":2213},[768],[575,2215,2217],{"className":2216},[772],[575,2218,2220],{"className":2219},[776],[575,2221,2223],{"className":2222,"style":755},[780],[575,2224,2225,2228],{"style":783},[575,2226],{"className":2227,"style":788},[787],[575,2229,2231],{"className":2230},[792,793,794,795],[575,2232,742],{"className":2233},[618,795]," ist falsch.",[2065,2236,2237,2238,2372,2373,2525],{},"Faktor und Summand verwechseln: ",[575,2239,2241,2275],{"className":2240},[578],[575,2242,2244],{"className":2243},[582],[584,2245,2246],{"xmlns":586},[588,2247,2248,2272],{},[591,2249,2250,2252,2254,2260,2266,2268,2270],{},[856,2251,1016],{"stretchy":1015},[731,2253,733],{},[735,2255,2256,2258],{},[594,2257,739],{},[731,2259,742],{},[735,2261,2262,2264],{},[856,2263,1021],{"stretchy":1015},[856,2265,1052],{"mathvariant":1050,"lspace":1051,"rspace":1051},[856,2267,1024],{},[731,2269,742],{},[594,2271,739],{},[598,2273,2274],{"encoding":600},"(5x^2)' = 2x",[575,2276,2278,2360],{"className":2277,"ariaHidden":605},[604],[575,2279,2281,2284,2287,2290,2319,2351,2354,2357],{"className":2280},[609],[575,2282],{"className":2283,"style":2118},[613],[575,2285,1016],{"className":2286},[1105],[575,2288,733],{"className":2289},[618],[575,2291,2293,2296],{"className":2292},[618],[575,2294,739],{"className":2295},[618,619],[575,2297,2299],{"className":2298},[768],[575,2300,2302],{"className":2301},[772],[575,2303,2305],{"className":2304},[776],[575,2306,2308],{"className":2307,"style":755},[780],[575,2309,2310,2313],{"style":783},[575,2311],{"className":2312,"style":788},[787],[575,2314,2316],{"className":2315},[792,793,794,795],[575,2317,742],{"className":2318},[618,795],[575,2320,2322,2325],{"className":2321},[1112],[575,2323,1021],{"className":2324},[1112],[575,2326,2328],{"className":2327},[768],[575,2329,2331],{"className":2330},[772],[575,2332,2334],{"className":2333},[776],[575,2335,2337],{"className":2336,"style":2172},[780],[575,2338,2339,2342],{"style":783},[575,2340],{"className":2341,"style":788},[787],[575,2343,2345],{"className":2344},[792,793,794,795],[575,2346,2348],{"className":2347},[618,795],[575,2349,1052],{"className":2350},[618,795],[575,2352],{"className":2353,"style":1116},[906],[575,2355,1024],{"className":2356},[1120],[575,2358],{"className":2359,"style":1116},[906],[575,2361,2363,2366,2369],{"className":2362},[609],[575,2364],{"className":2365,"style":829},[613],[575,2367,742],{"className":2368},[618],[575,2370,739],{"className":2371},[618,619]," ist falsch, ",[575,2374,2376,2412],{"className":2375},[578],[575,2377,2379],{"className":2378},[582],[584,2380,2381],{"xmlns":586},[588,2382,2383,2409],{},[591,2384,2385,2387,2393,2395,2397,2403,2405,2407],{},[856,2386,1016],{"stretchy":1015},[735,2388,2389,2391],{},[594,2390,739],{},[731,2392,742],{},[856,2394,858],{},[731,2396,733],{},[735,2398,2399,2401],{},[856,2400,1021],{"stretchy":1015},[856,2402,1052],{"mathvariant":1050,"lspace":1051,"rspace":1051},[856,2404,1024],{},[731,2406,742],{},[594,2408,739],{},[598,2410,2411],{"encoding":600},"(x^2 + 5)' = 2x",[575,2413,2415,2462,2513],{"className":2414,"ariaHidden":605},[604],[575,2416,2418,2421,2424,2453,2456,2459],{"className":2417},[609],[575,2419],{"className":2420,"style":2118},[613],[575,2422,1016],{"className":2423},[1105],[575,2425,2427,2430],{"className":2426},[618],[575,2428,739],{"className":2429},[618,619],[575,2431,2433],{"className":2432},[768],[575,2434,2436],{"className":2435},[772],[575,2437,2439],{"className":2438},[776],[575,2440,2442],{"className":2441,"style":755},[780],[575,2443,2444,2447],{"style":783},[575,2445],{"className":2446,"style":788},[787],[575,2448,2450],{"className":2449},[792,793,794,795],[575,2451,742],{"className":2452},[618,795],[575,2454],{"className":2455,"style":907},[906],[575,2457,858],{"className":2458},[911],[575,2460],{"className":2461,"style":907},[906],[575,2463,2465,2469,2472,2504,2507,2510],{"className":2464},[609],[575,2466],{"className":2467,"style":2468},[613],"height:1.0019em;vertical-align:-0.25em;",[575,2470,733],{"className":2471},[618],[575,2473,2475,2478],{"className":2474},[1112],[575,2476,1021],{"className":2477},[1112],[575,2479,2481],{"className":2480},[768],[575,2482,2484],{"className":2483},[772],[575,2485,2487],{"className":2486},[776],[575,2488,2490],{"className":2489,"style":2172},[780],[575,2491,2492,2495],{"style":783},[575,2493],{"className":2494,"style":788},[787],[575,2496,2498],{"className":2497},[792,793,794,795],[575,2499,2501],{"className":2500},[618,795],[575,2502,1052],{"className":2503},[618,795],[575,2505],{"className":2506,"style":1116},[906],[575,2508,1024],{"className":2509},[1120],[575,2511],{"className":2512,"style":1116},[906],[575,2514,2516,2519,2522],{"className":2515},[609],[575,2517],{"className":2518,"style":829},[613],[575,2520,742],{"className":2521},[618],[575,2523,739],{"className":2524},[618,619]," ist richtig.",[2065,2527,2528,2529,988],{},"Den Faktor selbst „ableiten“ und die Produktregel bemühen. Das geht zwar, liefert aber nur ",[575,2530,2532,2593],{"className":2531},[578],[575,2533,2535],{"className":2534},[582],[584,2536,2537],{"xmlns":586},[588,2538,2539,2590],{},[591,2540,2541,2543,2545,2548,2550,2552,2554,2556,2558,2560,2566,2568,2570,2572,2574,2576,2578,2584,2586,2588],{},[731,2542,1459],{},[856,2544,1065],{},[594,2546,2547],{},"g",[856,2549,1016],{"stretchy":1015},[594,2551,739],{},[856,2553,1021],{"stretchy":1015},[856,2555,858],{},[594,2557,596],{},[856,2559,1065],{},[735,2561,2562,2564],{},[594,2563,2547],{},[856,2565,1052],{"mathvariant":1050,"lspace":1051,"rspace":1051},[856,2567,1016],{"stretchy":1015},[594,2569,739],{},[856,2571,1021],{"stretchy":1015},[856,2573,1024],{},[594,2575,596],{},[856,2577,1065],{},[735,2579,2580,2582],{},[594,2581,2547],{},[856,2583,1052],{"mathvariant":1050,"lspace":1051,"rspace":1051},[856,2585,1016],{"stretchy":1015},[594,2587,739],{},[856,2589,1021],{"stretchy":1015},[598,2591,2592],{"encoding":600},"0 \\cdot g(x) + c \\cdot g'(x) = c \\cdot g'(x)",[575,2594,2596,2614,2641,2660,2716,2734],{"className":2595,"ariaHidden":605},[604],[575,2597,2599,2602,2605,2608,2611],{"className":2598},[609],[575,2600],{"className":2601,"style":829},[613],[575,2603,1459],{"className":2604},[618],[575,2606],{"className":2607,"style":907},[906],[575,2609,1065],{"className":2610},[911],[575,2612],{"className":2613,"style":907},[906],[575,2615,2617,2620,2623,2626,2629,2632,2635,2638],{"className":2616},[609],[575,2618],{"className":2619,"style":1097},[613],[575,2621,2547],{"className":2622,"style":651},[618,619],[575,2624,1016],{"className":2625},[1105],[575,2627,739],{"className":2628},[618,619],[575,2630,1021],{"className":2631},[1112],[575,2633],{"className":2634,"style":907},[906],[575,2636,858],{"className":2637},[911],[575,2639],{"className":2640,"style":907},[906],[575,2642,2644,2648,2651,2654,2657],{"className":2643},[609],[575,2645],{"className":2646,"style":2647},[613],"height:0.4445em;",[575,2649,596],{"className":2650},[618,619],[575,2652],{"className":2653,"style":907},[906],[575,2655,1065],{"className":2656},[911],[575,2658],{"className":2659,"style":907},[906],[575,2661,2663,2666,2698,2701,2704,2707,2710,2713],{"className":2662},[609],[575,2664],{"className":2665,"style":2468},[613],[575,2667,2669,2672],{"className":2668},[618],[575,2670,2547],{"className":2671,"style":651},[618,619],[575,2673,2675],{"className":2674},[768],[575,2676,2678],{"className":2677},[772],[575,2679,2681],{"className":2680},[776],[575,2682,2684],{"className":2683,"style":2172},[780],[575,2685,2686,2689],{"style":783},[575,2687],{"className":2688,"style":788},[787],[575,2690,2692],{"className":2691},[792,793,794,795],[575,2693,2695],{"className":2694},[618,795],[575,2696,1052],{"className":2697},[618,795],[575,2699,1016],{"className":2700},[1105],[575,2702,739],{"className":2703},[618,619],[575,2705,1021],{"className":2706},[1112],[575,2708],{"className":2709,"style":1116},[906],[575,2711,1024],{"className":2712},[1120],[575,2714],{"className":2715,"style":1116},[906],[575,2717,2719,2722,2725,2728,2731],{"className":2718},[609],[575,2720],{"className":2721,"style":2647},[613],[575,2723,596],{"className":2724},[618,619],[575,2726],{"className":2727,"style":907},[906],[575,2729,1065],{"className":2730},[911],[575,2732],{"className":2733,"style":907},[906],[575,2735,2737,2740,2772,2775,2778],{"className":2736},[609],[575,2738],{"className":2739,"style":2468},[613],[575,2741,2743,2746],{"className":2742},[618],[575,2744,2547],{"className":2745,"style":651},[618,619],[575,2747,2749],{"className":2748},[768],[575,2750,2752],{"className":2751},[772],[575,2753,2755],{"className":2754},[776],[575,2756,2758],{"className":2757,"style":2172},[780],[575,2759,2760,2763],{"style":783},[575,2761],{"className":2762,"style":788},[787],[575,2764,2766],{"className":2765},[792,793,794,795],[575,2767,2769],{"className":2768},[618,795],[575,2770,1052],{"className":2771},[618,795],[575,2773,1016],{"className":2774},[1105],[575,2776,739],{"className":2777},[618,619],[575,2779,1021],{"className":2780},[1112],{"title":2782,"searchDepth":2783,"depth":2783,"links":2784},"",2,[2785,2786,2787],{"id":560,"depth":2783,"text":561},{"id":991,"depth":2783,"text":992},{"id":2059,"depth":2783,"text":2060},[2789,2791,2794,2797,2800],{"lhs":2790,"rhs":1459},"round((k*(x+1e-5)^3 - k*(x-1e-5)^3)\u002F(2e-5) - k*3*x^2, 6)",{"lhs":2792,"rhs":2793},"4*3*x^2","12*x^2",{"lhs":2795,"rhs":2796},"-0.5*2*x","-x",{"lhs":2798,"rhs":2799},"((x+h)\u002F3 - x\u002F3)\u002Fh","1\u002F3",{"lhs":2801,"rhs":2802},"5*2*x","10*x","c \\in \\mathbb{R},\\ g \\text{ differenzierbar}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>c\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">c \\in \\mathbb{R},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8833em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>g\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">g\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">differenzierbar\u003C\u002Fspan>",null,"md","constant-factor-rule","f(x) = c \\cdot g(x) \\;\\Rightarrow\\; f'(x) = c \\cdot g'(x)","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>f\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>g\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>⇒\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmi>f\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmsup>\u003Cmi>g\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f(x) = c \\cdot g(x) \\;\\Rightarrow\\; f'(x) = c \\cdot g'(x)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4445em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⇒\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.0519em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8019em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4445em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.0519em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8019em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fcalculus\u002Ffaktorregel",[],[2816,2817],"power-rule","sum-rule",{"title":548,"description":2782},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Ffaktorregel","Ein konstanter Faktor bleibt beim Ableiten einfach stehen, abgeleitet wird nur die Funktion dahinter.",[2822,2823,2824,2825],"ableitung","ableitungsregel","faktor","konstante",[2827,2830,2833],{"symbol":596,"meaning":2828,"symbolHtml":2829},"konstanter Faktor, eine feste Zahl","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">c\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":2547,"meaning":2831,"symbolHtml":2832},"differenzierbare Funktion","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>g\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">g\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":2834,"meaning":2835,"symbolHtml":2836},"f'(x), g'(x)","Ableitungen von f und g","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>f\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsup>\u003Cmi>g\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f'(x), g'(x)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.0019em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","IN7GpOilT-gXIDUNlsjdG-gu4c4MYnGujgb216yYLNc",[2839,2842],{"stem":2840,"title":2841,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fpotenzregel","Potenzregel",{"stem":2843,"title":2844,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fsummenregel","Summenregel",1793824281531]