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\\frac{1}{x}",[569,1132,1134,1208],{"className":1133,"ariaHidden":607},[606],[569,1135,1137,1141,1199,1202,1205],{"className":1136},[611],[569,1138],{"className":1139,"style":1140},[615],"height:1.1418em;vertical-align:-0.25em;",[569,1142,1145,1171],{"className":1143},[1144],"minner",[569,1146,1148,1153,1156,1159,1162,1165,1168],{"className":1147},[1144],[569,1149,1101],{"className":1150,"style":1152},[730,1151],"delimcenter","top:0em;",[569,1154,590],{"className":1155},[620],[569,1157],{"className":1158,"style":625},[624],[569,1160,1108],{"className":1161},[629],[569,1163,597],{"className":1164},[629,630],[569,1166,1108],{"className":1167},[629],[569,1169,1115],{"className":1170,"style":1152},[813,1151],[569,1172,1174],{"className":1173},[958],[569,1175,1177],{"className":1176},[738],[569,1178,1180],{"className":1179},[743],[569,1181,1184],{"className":1182,"style":1183},[747],"height:0.8918em;",[569,1185,1187,1190],{"style":1186},"top:-3.2029em;margin-right:0.05em;",[569,1188],{"className":1189,"style":974},[755],[569,1191,1193],{"className":1192},[760,761,762,763],[569,1194,1196],{"className":1195},[629,763],[569,1197,1119],{"className":1198},[629,763],[569,1200],{"className":1201,"style":672},[624],[569,1203,1064],{"className":1204},[676],[569,1206],{"className":1207,"style":672},[624],[569,1209,1211,1215],{"className":1210},[611],[569,1212],{"className":1213,"style":1214},[615],"height:2.0074em;vertical-align:-0.686em;",[569,1216,1218,1221,1278],{"className":1217},[629],[569,1219],{"className":1220},[730,731],[569,1222,1224],{"className":1223},[704],[569,1225,1227,1269],{"className":1226},[738,739],[569,1228,1230,1266],{"className":1229},[743],[569,1231,1234,1246,1254],{"className":1232,"style":1233},[747],"height:1.3214em;",[569,1235,1237,1240],{"style":1236},"top:-2.314em;",[569,1238],{"className":1239,"style":756},[755],[569,1241,1243],{"className":1242},[629],[569,1244,597],{"className":1245},[629,630],[569,1247,1248,1251],{"style":772},[569,1249],{"className":1250,"style":756},[755],[569,1252],{"className":1253,"style":780},[779],[569,1255,1257,1260],{"style":1256},"top:-3.677em;",[569,1258],{"className":1259,"style":756},[755],[569,1261,1263],{"className":1262},[629],[569,1264,708],{"className":1265},[629],[569,1267,800],{"className":1268},[799],[569,1270,1272],{"className":1271},[743],[569,1273,1276],{"className":1274,"style":1275},[747],"height:0.686em;",[569,1277],{},[569,1279],{"className":1280},[813,731],[565,1282,1283,1284,1336,1337,1421,1422,1691],{},"Für einen Logarithmus zu einer anderen Basis ",[569,1285,1287,1306],{"className":1286},[572],[569,1288,1290],{"className":1289},[576],[578,1291,1292],{"xmlns":580},[582,1293,1294,1303],{},[585,1295,1296,1299,1301],{},[588,1297,1298],{},"a",[592,1300,648],{},[650,1302,652],{},[599,1304,1305],{"encoding":601},"a > 0",[569,1307,1309,1327],{"className":1308,"ariaHidden":607},[606],[569,1310,1312,1315,1318,1321,1324],{"className":1311},[611],[569,1313],{"className":1314,"style":665},[615],[569,1316,1298],{"className":1317},[629,630],[569,1319],{"className":1320,"style":672},[624],[569,1322,648],{"className":1323},[676],[569,1325],{"className":1326,"style":672},[624],[569,1328,1330,1333],{"className":1329},[611],[569,1331],{"className":1332,"style":686},[615],[569,1334,652],{"className":1335},[629],", ",[569,1338,1340,1358],{"className":1339},[572],[569,1341,1343],{"className":1342},[576],[578,1344,1345],{"xmlns":580},[582,1346,1347,1355],{},[585,1348,1349,1351,1353],{},[588,1350,1298],{},[592,1352,999],{"mathvariant":998},[650,1354,708],{},[599,1356,1357],{"encoding":601},"a \\neq 1",[569,1359,1361,1412],{"className":1360,"ariaHidden":607},[606],[569,1362,1364,1367,1370,1373,1409],{"className":1363},[611],[569,1365],{"className":1366,"style":1014},[615],[569,1368,1298],{"className":1369},[629,630],[569,1371],{"className":1372,"style":672},[624],[569,1374,1376,1403,1406],{"className":1375},[676],[569,1377,1379],{"className":1378},[676],[569,1380,1382],{"className":1381},[629,1030],[569,1383,1385],{"className":1384},[1034],[569,1386,1388,1391,1400],{"className":1387},[1038],[569,1389],{"className":1390,"style":1014},[615],[569,1392,1394],{"className":1393},[1045],[569,1395,1397],{"className":1396},[629],[569,1398,1052],{"className":1399},[676],[569,1401],{"className":1402},[1056],[569,1404],{"className":1405},[624,1060],[569,1407,1064],{"className":1408},[676],[569,1410],{"className":1411,"style":672},[624],[569,1413,1415,1418],{"className":1414},[611],[569,1416],{"className":1417,"style":686},[615],[569,1419,708],{"className":1420},[629]," folgt mit dem Basiswechsel: ",[569,1423,1425,1482],{"className":1424},[572],[569,1426,1428],{"className":1427},[576],[578,1429,1430],{"xmlns":580},[582,1431,1432,1479],{},[585,1433,1434,1458,1460],{},[929,1435,1436,1456],{},[585,1437,1438,1440,1452,1454],{},[592,1439,1101],{"fence":607},[1441,1442,1443,1450],"msub",{},[585,1444,1445,1448],{},[588,1446,1447],{},"log",[592,1449,594],{},[588,1451,1298],{},[588,1453,597],{},[592,1455,1115],{"fence":607},[592,1457,1119],{"mathvariant":998,"lspace":1118,"rspace":1118},[592,1459,1064],{},[704,1461,1462,1464],{},[650,1463,708],{},[585,1465,1466,1468,1470,1472,1475,1477],{},[588,1467,597],{},[588,1469,590],{},[592,1471,594],{},[592,1473,1101],{"stretchy":1474},"false",[588,1476,1298],{},[592,1478,1115],{"stretchy":1474},[599,1480,1481],{"encoding":601},"\\left(\\log_a x\\right)' = \\frac{1}{x \\ln(a)}",[569,1483,1485,1592],{"className":1484,"ariaHidden":607},[606],[569,1486,1488,1491,1583,1586,1589],{"className":1487},[611],[569,1489],{"className":1490,"style":1140},[615],[569,1492,1494,1557],{"className":1493},[1144],[569,1495,1497,1500,1548,1551,1554],{"className":1496},[1144],[569,1498,1101],{"className":1499,"style":1152},[730,1151],[569,1501,1503,1511],{"className":1502},[620],[569,1504,1506,1507],{"className":1505},[620],"lo",[569,1508,1510],{"style":1509},"margin-right:0.0139em;","g",[569,1512,1514],{"className":1513},[958],[569,1515,1517,1539],{"className":1516},[738,739],[569,1518,1520,1536],{"className":1519},[743],[569,1521,1524],{"className":1522,"style":1523},[747],"height:0.0573em;",[569,1525,1527,1530],{"style":1526},"top:-2.4559em;margin-right:0.05em;",[569,1528],{"className":1529,"style":974},[755],[569,1531,1533],{"className":1532},[760,761,762,763],[569,1534,1298],{"className":1535},[629,630,763],[569,1537,800],{"className":1538},[799],[569,1540,1542],{"className":1541},[743],[569,1543,1546],{"className":1544,"style":1545},[747],"height:0.2441em;",[569,1547],{},[569,1549],{"className":1550,"style":625},[624],[569,1552,597],{"className":1553},[629,630],[569,1555,1115],{"className":1556,"style":1152},[813,1151],[569,1558,1560],{"className":1559},[958],[569,1561,1563],{"className":1562},[738],[569,1564,1566],{"className":1565},[743],[569,1567,1569],{"className":1568,"style":1183},[747],[569,1570,1571,1574],{"style":1186},[569,1572],{"className":1573,"style":974},[755],[569,1575,1577],{"className":1576},[760,761,762,763],[569,1578,1580],{"className":1579},[629,763],[569,1581,1119],{"className":1582},[629,763],[569,1584],{"className":1585,"style":672},[624],[569,1587,1064],{"className":1588},[676],[569,1590],{"className":1591,"style":672},[624],[569,1593,1595,1599],{"className":1594},[611],[569,1596],{"className":1597,"style":1598},[615],"height:1.3651em;vertical-align:-0.52em;",[569,1600,1602,1605,1688],{"className":1601},[629],[569,1603],{"className":1604},[730,731],[569,1606,1608],{"className":1607},[704],[569,1609,1611,1679],{"className":1610},[738,739],[569,1612,1614,1676],{"className":1613},[743],[569,1615,1617,1654,1662],{"className":1616,"style":748},[747],[569,1618,1619,1622],{"style":751},[569,1620],{"className":1621,"style":756},[755],[569,1623,1625],{"className":1624},[760,761,762,763],[569,1626,1628,1631,1635,1645,1648,1651],{"className":1627},[629,763],[569,1629,597],{"className":1630},[629,630,763],[569,1632],{"className":1633,"style":1634},[624,763],"margin-right:0.1952em;",[569,1636,1638,1642],{"className":1637},[620,763],[569,1639,1641],{"className":1640},[763],"l",[569,1643,935],{"className":1644},[763],[569,1646,1101],{"className":1647},[730,763],[569,1649,1298],{"className":1650},[629,630,763],[569,1652,1115],{"className":1653},[813,763],[569,1655,1656,1659],{"style":772},[569,1657],{"className":1658,"style":756},[755],[569,1660],{"className":1661,"style":780},[779],[569,1663,1664,1667],{"style":783},[569,1665],{"className":1666,"style":756},[755],[569,1668,1670],{"className":1669},[760,761,762,763],[569,1671,1673],{"className":1672},[629,763],[569,1674,708],{"className":1675},[629,763],[569,1677,800],{"className":1678},[799],[569,1680,1682],{"className":1681},[743],[569,1683,1686],{"className":1684,"style":1685},[747],"height:0.52em;",[569,1687],{},[569,1689],{"className":1690},[813,731],".",[560,1693,1695],{"id":1694},"herleitung","Herleitung",[565,1697,1698,1729,1730,1835],{},[569,1699,1701,1717],{"className":1700},[572],[569,1702,1704],{"className":1703},[576],[578,1705,1706],{"xmlns":580},[582,1707,1708,1714],{},[585,1709,1710,1712],{},[588,1711,590],{},[592,1713,594],{},[599,1715,1716],{"encoding":601},"\\ln",[569,1718,1720],{"className":1719,"ariaHidden":607},[606],[569,1721,1723,1726],{"className":1722},[611],[569,1724],{"className":1725,"style":616},[615],[569,1727,590],{"className":1728},[620]," ist die Umkehrfunktion der e-Funktion, also gilt ",[569,1731,1733,1762],{"className":1732},[572],[569,1734,1736],{"className":1735},[576],[578,1737,1738],{"xmlns":580},[582,1739,1740,1759],{},[585,1741,1742,1755,1757],{},[929,1743,1744,1747],{},[588,1745,1746],{},"e",[585,1748,1749,1751,1753],{},[588,1750,590],{},[592,1752,594],{},[588,1754,597],{},[592,1756,1064],{},[588,1758,597],{},[599,1760,1761],{"encoding":601},"e^{\\ln x} = x",[569,1763,1765,1825],{"className":1764,"ariaHidden":607},[606],[569,1766,1768,1772,1816,1819,1822],{"className":1767},[611],[569,1769],{"className":1770,"style":1771},[615],"height:0.8491em;",[569,1773,1775,1778],{"className":1774},[629],[569,1776,1746],{"className":1777},[629,630],[569,1779,1781],{"className":1780},[958],[569,1782,1784],{"className":1783},[738],[569,1785,1787],{"className":1786},[743],[569,1788,1790],{"className":1789,"style":1771},[747],[569,1791,1792,1795],{"style":970},[569,1793],{"className":1794,"style":974},[755],[569,1796,1798],{"className":1797},[760,761,762,763],[569,1799,1801,1810,1813],{"className":1800},[629,763],[569,1802,1804,1807],{"className":1803},[620,763],[569,1805,1641],{"className":1806},[763],[569,1808,935],{"className":1809},[763],[569,1811],{"className":1812,"style":1634},[624,763],[569,1814,597],{"className":1815},[629,630,763],[569,1817],{"className":1818,"style":672},[624],[569,1820,1064],{"className":1821},[676],[569,1823],{"className":1824,"style":672},[624],[569,1826,1828,1832],{"className":1827},[611],[569,1829],{"className":1830,"style":1831},[615],"height:0.4306em;",[569,1833,597],{"className":1834},[629,630],". Leitet man beide Seiten mit der Kettenregel ab, erhält man",[569,1837,1839],{"className":1838},[1081],[569,1840,1842,1935],{"className":1841},[572],[569,1843,1845],{"className":1844},[576],[578,1846,1847],{"xmlns":580,"display":1090},[582,1848,1849,1932],{},[585,1850,1851,1863,1866,1868,1870,1872,1874,1880,1882,1884,1887,1890,1892,1894,1896,1898,1900,1906,1908,1924,1926],{},[929,1852,1853,1855],{},[588,1854,1746],{},[585,1856,1857,1859,1861],{},[588,1858,590],{},[592,1860,594],{},[588,1862,597],{},[592,1864,1865],{},"⋅",[592,1867,1101],{"stretchy":1474},[588,1869,590],{},[592,1871,594],{},[588,1873,597],{},[929,1875,1876,1878],{},[592,1877,1115],{"stretchy":1474},[592,1879,1119],{"mathvariant":998,"lspace":1118,"rspace":1118},[592,1881,1064],{},[650,1883,708],{},[624,1885],{"width":1886},"1em",[592,1888,1889],{},"⇒",[624,1891],{"width":1886},[592,1893,1101],{"stretchy":1474},[588,1895,590],{},[592,1897,594],{},[588,1899,597],{},[929,1901,1902,1904],{},[592,1903,1115],{"stretchy":1474},[592,1905,1119],{"mathvariant":998,"lspace":1118,"rspace":1118},[592,1907,1064],{},[704,1909,1910,1912],{},[650,1911,708],{},[929,1913,1914,1916],{},[588,1915,1746],{},[585,1917,1918,1920,1922],{},[588,1919,590],{},[592,1921,594],{},[588,1923,597],{},[592,1925,1064],{},[704,1927,1928,1930],{},[650,1929,708],{},[588,1931,597],{},[599,1933,1934],{"encoding":601},"e^{\\ln x} \\cdot (\\ln x)' = 1 \\quad\\Rightarrow\\quad (\\ln x)' = \\frac{1}{e^{\\ln x}} = \\frac{1}{x}",[569,1936,1938,2001,2062,2087,2146,2266],{"className":1937,"ariaHidden":607},[606],[569,1939,1941,1945,1990,1994,1998],{"className":1940},[611],[569,1942],{"className":1943,"style":1944},[615],"height:0.8991em;",[569,1946,1948,1951],{"className":1947},[629],[569,1949,1746],{"className":1950},[629,630],[569,1952,1954],{"className":1953},[958],[569,1955,1957],{"className":1956},[738],[569,1958,1960],{"className":1959},[743],[569,1961,1963],{"className":1962,"style":1944},[747],[569,1964,1966,1969],{"style":1965},"top:-3.113em;margin-right:0.05em;",[569,1967],{"className":1968,"style":974},[755],[569,1970,1972],{"className":1971},[760,761,762,763],[569,1973,1975,1984,1987],{"className":1974},[629,763],[569,1976,1978,1981],{"className":1977},[620,763],[569,1979,1641],{"className":1980},[763],[569,1982,935],{"className":1983},[763],[569,1985],{"className":1986,"style":1634},[624,763],[569,1988,597],{"className":1989},[629,630,763],[569,1991],{"className":1992,"style":1993},[624],"margin-right:0.2222em;",[569,1995,1865],{"className":1996},[1997],"mbin",[569,1999],{"className":2000,"style":1993},[624],[569,2002,2004,2008,2011,2014,2017,2020,2053,2056,2059],{"className":2003},[611],[569,2005],{"className":2006,"style":2007},[615],"height:1.0519em;vertical-align:-0.25em;",[569,2009,1101],{"className":2010},[730],[569,2012,590],{"className":2013},[620],[569,2015],{"className":2016,"style":625},[624],[569,2018,597],{"className":2019},[629,630],[569,2021,2023,2026],{"className":2022},[813],[569,2024,1115],{"className":2025},[813],[569,2027,2029],{"className":2028},[958],[569,2030,2032],{"className":2031},[738],[569,2033,2035],{"className":2034},[743],[569,2036,2039],{"className":2037,"style":2038},[747],"height:0.8019em;",[569,2040,2041,2044],{"style":1965},[569,2042],{"className":2043,"style":974},[755],[569,2045,2047],{"className":2046},[760,761,762,763],[569,2048,2050],{"className":2049},[629,763],[569,2051,1119],{"className":2052},[629,763],[569,2054],{"className":2055,"style":672},[624],[569,2057,1064],{"className":2058},[676],[569,2060],{"className":2061,"style":672},[624],[569,2063,2065,2068,2071,2075,2078,2081,2084],{"className":2064},[611],[569,2066],{"className":2067,"style":686},[615],[569,2069,708],{"className":2070},[629],[569,2072],{"className":2073,"style":2074},[624],"margin-right:1em;",[569,2076],{"className":2077,"style":672},[624],[569,2079,1889],{"className":2080},[676],[569,2082],{"className":2083,"style":2074},[624],[569,2085],{"className":2086,"style":672},[624],[569,2088,2090,2093,2096,2099,2102,2105,2137,2140,2143],{"className":2089},[611],[569,2091],{"className":2092,"style":2007},[615],[569,2094,1101],{"className":2095},[730],[569,2097,590],{"className":2098},[620],[569,2100],{"className":2101,"style":625},[624],[569,2103,597],{"className":2104},[629,630],[569,2106,2108,2111],{"className":2107},[813],[569,2109,1115],{"className":2110},[813],[569,2112,2114],{"className":2113},[958],[569,2115,2117],{"className":2116},[738],[569,2118,2120],{"className":2119},[743],[569,2121,2123],{"className":2122,"style":2038},[747],[569,2124,2125,2128],{"style":1965},[569,2126],{"className":2127,"style":974},[755],[569,2129,2131],{"className":2130},[760,761,762,763],[569,2132,2134],{"className":2133},[629,763],[569,2135,1119],{"className":2136},[629,763],[569,2138],{"className":2139,"style":672},[624],[569,2141,1064],{"className":2142},[676],[569,2144],{"className":2145,"style":672},[624],[569,2147,2149,2152,2257,2260,2263],{"className":2148},[611],[569,2150],{"className":2151,"style":1214},[615],[569,2153,2155,2158,2254],{"className":2154},[629],[569,2156],{"className":2157},[730,731],[569,2159,2161],{"className":2160},[704],[569,2162,2164,2246],{"className":2163},[738,739],[569,2165,2167,2243],{"className":2166},[743],[569,2168,2170,2224,2232],{"className":2169,"style":1233},[747],[569,2171,2172,2175],{"style":1236},[569,2173],{"className":2174,"style":756},[755],[569,2176,2178],{"className":2177},[629],[569,2179,2181,2184],{"className":2180},[629],[569,2182,1746],{"className":2183},[629,630],[569,2185,2187],{"className":2186},[958],[569,2188,2190],{"className":2189},[738],[569,2191,2193],{"className":2192},[743],[569,2194,2197],{"className":2195,"style":2196},[747],"height:0.7751em;",[569,2198,2200,2203],{"style":2199},"top:-2.989em;margin-right:0.05em;",[569,2201],{"className":2202,"style":974},[755],[569,2204,2206],{"className":2205},[760,761,762,763],[569,2207,2209,2218,2221],{"className":2208},[629,763],[569,2210,2212,2215],{"className":2211},[620,763],[569,2213,1641],{"className":2214},[763],[569,2216,935],{"className":2217},[763],[569,2219],{"className":2220,"style":1634},[624,763],[569,2222,597],{"className":2223},[629,630,763],[569,2225,2226,2229],{"style":772},[569,2227],{"className":2228,"style":756},[755],[569,2230],{"className":2231,"style":780},[779],[569,2233,2234,2237],{"style":1256},[569,2235],{"className":2236,"style":756},[755],[569,2238,2240],{"className":2239},[629],[569,2241,708],{"className":2242},[629],[569,2244,800],{"className":2245},[799],[569,2247,2249],{"className":2248},[743],[569,2250,2252],{"className":2251,"style":1275},[747],[569,2253],{},[569,2255],{"className":2256},[813,731],[569,2258],{"className":2259,"style":672},[624],[569,2261,1064],{"className":2262},[676],[569,2264],{"className":2265,"style":672},[624],[569,2267,2269,2272],{"className":2268},[611],[569,2270],{"className":2271,"style":1214},[615],[569,2273,2275,2278,2331],{"className":2274},[629],[569,2276],{"className":2277},[730,731],[569,2279,2281],{"className":2280},[704],[569,2282,2284,2323],{"className":2283},[738,739],[569,2285,2287,2320],{"className":2286},[743],[569,2288,2290,2301,2309],{"className":2289,"style":1233},[747],[569,2291,2292,2295],{"style":1236},[569,2293],{"className":2294,"style":756},[755],[569,2296,2298],{"className":2297},[629],[569,2299,597],{"className":2300},[629,630],[569,2302,2303,2306],{"style":772},[569,2304],{"className":2305,"style":756},[755],[569,2307],{"className":2308,"style":780},[779],[569,2310,2311,2314],{"style":1256},[569,2312],{"className":2313,"style":756},[755],[569,2315,2317],{"className":2316},[629],[569,2318,708],{"className":2319},[629],[569,2321,800],{"className":2322},[799],[569,2324,2326],{"className":2325},[743],[569,2327,2329],{"className":2328,"style":1275},[747],[569,2330],{},[569,2332],{"className":2333},[813,731],[560,2335,2337],{"id":2336},"beispiele","Beispiele",[565,2339,2340,2341,2460,2461,2500],{},"Mit dem Logarithmusgesetz ist ",[569,2342,2344,2386],{"className":2343},[572],[569,2345,2347],{"className":2346},[576],[578,2348,2349],{"xmlns":580},[582,2350,2351,2383],{},[585,2352,2353,2355,2357,2359,2362,2364,2366,2368,2370,2372,2374,2377,2379,2381],{},[588,2354,590],{},[592,2356,594],{},[592,2358,1101],{"stretchy":1474},[650,2360,2361],{},"3",[588,2363,597],{},[592,2365,1115],{"stretchy":1474},[592,2367,1064],{},[588,2369,590],{},[592,2371,594],{},[650,2373,2361],{},[592,2375,2376],{},"+",[588,2378,590],{},[592,2380,594],{},[588,2382,597],{},[599,2384,2385],{"encoding":601},"\\ln(3x) = \\ln 3 + \\ln x",[569,2387,2389,2420,2445],{"className":2388,"ariaHidden":607},[606],[569,2390,2392,2396,2399,2402,2405,2408,2411,2414,2417],{"className":2391},[611],[569,2393],{"className":2394,"style":2395},[615],"height:1em;vertical-align:-0.25em;",[569,2397,590],{"className":2398},[620],[569,2400,1101],{"className":2401},[730],[569,2403,2361],{"className":2404},[629],[569,2406,597],{"className":2407},[629,630],[569,2409,1115],{"className":2410},[813],[569,2412],{"className":2413,"style":672},[624],[569,2415,1064],{"className":2416},[676],[569,2418],{"className":2419,"style":672},[624],[569,2421,2423,2427,2430,2433,2436,2439,2442],{"className":2422},[611],[569,2424],{"className":2425,"style":2426},[615],"height:0.7778em;vertical-align:-0.0833em;",[569,2428,590],{"className":2429},[620],[569,2431],{"className":2432,"style":625},[624],[569,2434,2361],{"className":2435},[629],[569,2437],{"className":2438,"style":1993},[624],[569,2440,2376],{"className":2441},[1997],[569,2443],{"className":2444,"style":1993},[624],[569,2446,2448,2451,2454,2457],{"className":2447},[611],[569,2449],{"className":2450,"style":616},[615],[569,2452,590],{"className":2453},[620],[569,2455],{"className":2456,"style":625},[624],[569,2458,597],{"className":2459},[629,630],". Der Summand ",[569,2462,2464,2482],{"className":2463},[572],[569,2465,2467],{"className":2466},[576],[578,2468,2469],{"xmlns":580},[582,2470,2471,2479],{},[585,2472,2473,2475,2477],{},[588,2474,590],{},[592,2476,594],{},[650,2478,2361],{},[599,2480,2481],{"encoding":601},"\\ln 3",[569,2483,2485],{"className":2484,"ariaHidden":607},[606],[569,2486,2488,2491,2494,2497],{"className":2487},[611],[569,2489],{"className":2490,"style":616},[615],[569,2492,590],{"className":2493},[620],[569,2495],{"className":2496,"style":625},[624],[569,2498,2361],{"className":2499},[629]," ist konstant:",[569,2502,2504],{"className":2503},[1081],[569,2505,2507,2570],{"className":2506},[572],[569,2508,2510],{"className":2509},[576],[578,2511,2512],{"xmlns":580,"display":1090},[582,2513,2514,2567],{},[585,2515,2516,2519,2521,2523,2525,2527,2529,2531,2533,2535,2537,2539,2543,2545,2547,2553,2555,2557,2559,2561],{},[588,2517,2518],{},"f",[592,2520,1101],{"stretchy":1474},[588,2522,597],{},[592,2524,1115],{"stretchy":1474},[592,2526,1064],{},[588,2528,590],{},[592,2530,594],{},[592,2532,1101],{"stretchy":1474},[650,2534,2361],{},[588,2536,597],{},[592,2538,1115],{"stretchy":1474},[2540,2541,2542],"mtext",{},"  ",[592,2544,1889],{},[2540,2546,2542],{},[929,2548,2549,2551],{},[588,2550,2518],{},[592,2552,1119],{"mathvariant":998,"lspace":1118,"rspace":1118},[592,2554,1101],{"stretchy":1474},[588,2556,597],{},[592,2558,1115],{"stretchy":1474},[592,2560,1064],{},[704,2562,2563,2565],{},[650,2564,708],{},[588,2566,597],{},[599,2568,2569],{"encoding":601},"f(x) = \\ln(3x) \\;\\Rightarrow\\; f'(x) = 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gleiche Ergebnis liefert die Kettenregel: ",[569,2765,2767,2801],{"className":2766},[572],[569,2768,2770],{"className":2769},[576],[578,2771,2772],{"xmlns":580},[582,2773,2774,2798],{},[585,2775,2776,2786,2788,2790,2792],{},[704,2777,2778,2780],{},[650,2779,708],{},[585,2781,2782,2784],{},[650,2783,2361],{},[588,2785,597],{},[592,2787,1865],{},[650,2789,2361],{},[592,2791,1064],{},[704,2793,2794,2796],{},[650,2795,708],{},[588,2797,597],{},[599,2799,2800],{"encoding":601},"\\frac{1}{3x} \\cdot 3 = 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= \\ln(x^2 + 1) \\;\\Rightarrow\\; f'(x) = \\frac{2x}{x^2 + 1}, \\quad f'(1) = \\frac{2}{2} = 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",[569,3585,3587,3609],{"className":3586},[572],[569,3588,3590],{"className":3589},[576],[578,3591,3592],{"xmlns":580},[582,3593,3594,3606],{},[585,3595,3596],{},[704,3597,3598,3600],{},[650,3599,708],{},[585,3601,3602,3604],{},[650,3603,2361],{},[588,3605,597],{},[599,3607,3608],{"encoding":601},"\\frac{1}{3x}",[569,3610,3612],{"className":3611,"ariaHidden":607},[606],[569,3613,3615,3618],{"className":3614},[611],[569,3616],{"className":3617,"style":723},[615],[569,3619,3621,3624,3686],{"className":3620},[629],[569,3622],{"className":3623},[730,731],[569,3625,3627],{"className":3626},[704],[569,3628,3630,3678],{"className":3629},[738,739],[569,3631,3633,3675],{"className":3632},[743],[569,3634,3636,3653,3661],{"className":3635,"style":748},[747],[569,3637,3638,3641],{"style":751},[569,3639],{"className":3640,"style":756},[755],[569,3642,3644],{"className":3643},[760,761,762,763],[569,3645,3647,3650],{"className":3646},[629,763],[569,3648,2361],{"className":3649},[629,763],[569,3651,597],{"className":3652},[629,630,763],[569,3654,3655,3658],{"style":772},[569,3656],{"className":3657,"style":756},[755],[569,3659],{"className":3660,"style":780},[779],[569,3662,3663,3666],{"style":783},[569,3664],{"className":3665,"style":756},[755],[569,3667,3669],{"className":3668},[760,761,762,763],[569,3670,3672],{"className":3671},[629,763],[569,3673,708],{"className":3674},[629,763],[569,3676,800],{"className":3677},[799],[569,3679,3681],{"className":3680},[743],[569,3682,3684],{"className":3683,"style":807},[747],[569,3685],{},[569,3687],{"className":3688},[813,731]," ist falsch, richtig ist ",[569,3691,3693,3710],{"className":3692},[572],[569,3694,3696],{"className":3695},[576],[578,3697,3698],{"xmlns":580},[582,3699,3700,3708],{},[585,3701,3702],{},[704,3703,3704,3706],{},[650,3705,708],{},[588,3707,597],{},[599,3709,713],{"encoding":601},[569,3711,3713],{"className":3712,"ariaHidden":607},[606],[569,3714,3716,3719],{"className":3715},[611],[569,3717],{"className":3718,"style":723},[615],[569,3720,3722,3725,3784],{"className":3721},[629],[569,3723],{"className":3724},[730,731],[569,3726,3728],{"className":3727},[704],[569,3729,3731,3776],{"className":3730},[738,739],[569,3732,3734,3773],{"className":3733},[743],[569,3735,3737,3751,3759],{"className":3736,"style":748},[747],[569,3738,3739,3742],{"style":751},[569,3740],{"className":3741,"style":756},[755],[569,3743,3745],{"className":3744},[760,761,762,763],[569,3746,3748],{"className":3747},[629,763],[569,3749,597],{"className":3750},[629,630,763],[569,3752,3753,3756],{"style":772},[569,3754],{"className":3755,"style":756},[755],[569,3757],{"className":3758,"style":780},[779],[569,3760,3761,3764],{"style":783},[569,3762],{"className":3763,"style":756},[755],[569,3765,3767],{"className":3766},[760,761,762,763],[569,3768,3770],{"className":3769},[629,763],[569,3771,708],{"className":3772},[629,763],[569,3774,800],{"className":3775},[799],[569,3777,3779],{"className":3778},[743],[569,3780,3782],{"className":3781,"style":807},[747],[569,3783],{},[569,3785],{"className":3786},[813,731],[3529,3788,3789,3790,3843,3844,3882],{},"Die Ableitung für ",[569,3791,3793,3812],{"className":3792},[572],[569,3794,3796],{"className":3795},[576],[578,3797,3798],{"xmlns":580},[582,3799,3800,3809],{},[585,3801,3802,3804,3807],{},[588,3803,597],{},[592,3805,3806],{},"≤",[650,3808,652],{},[599,3810,3811],{"encoding":601},"x \\leq 0",[569,3813,3815,3834],{"className":3814,"ariaHidden":607},[606],[569,3816,3818,3822,3825,3828,3831],{"className":3817},[611],[569,3819],{"className":3820,"style":3821},[615],"height:0.7719em;vertical-align:-0.136em;",[569,3823,597],{"className":3824},[629,630],[569,3826],{"className":3827,"style":672},[624],[569,3829,3806],{"className":3830},[676],[569,3832],{"className":3833,"style":672},[624],[569,3835,3837,3840],{"className":3836},[611],[569,3838],{"className":3839,"style":686},[615],[569,3841,652],{"className":3842},[629]," verwenden. Dort ist ",[569,3845,3847,3864],{"className":3846},[572],[569,3848,3850],{"className":3849},[576],[578,3851,3852],{"xmlns":580},[582,3853,3854,3862],{},[585,3855,3856,3858,3860],{},[588,3857,590],{},[592,3859,594],{},[588,3861,597],{},[599,3863,602],{"encoding":601},[569,3865,3867],{"className":3866,"ariaHidden":607},[606],[569,3868,3870,3873,3876,3879],{"className":3869},[611],[569,3871],{"className":3872,"style":616},[615],[569,3874,590],{"className":3875},[620],[569,3877],{"className":3878,"style":625},[624],[569,3880,597],{"className":3881},[629,630]," gar nicht definiert.",[3529,3884,3885,4078,4079,4107,4108,1691],{},[569,3886,3888,3928],{"className":3887},[572],[569,3889,3891],{"className":3890},[576],[578,3892,3893],{"xmlns":580},[582,3894,3895,3925],{},[585,3896,3897,3899,3901,3903,3905,3911,3913],{},[592,3898,1101],{"stretchy":1474},[588,3900,590],{},[592,3902,594],{},[588,3904,597],{},[929,3906,3907,3909],{},[592,3908,1115],{"stretchy":1474},[592,3910,1119],{"mathvariant":998,"lspace":1118,"rspace":1118},[592,3912,1064],{},[704,3914,3915,3917],{},[650,3916,708],{},[585,3918,3919,3921,3923],{},[588,3920,590],{},[592,3922,594],{},[588,3924,597],{},[599,3926,3927],{"encoding":601},"(\\ln x)' = \\frac{1}{\\ln x}",[569,3929,3931,3992],{"className":3930,"ariaHidden":607},[606],[569,3932,3934,3938,3941,3944,3947,3950,3983,3986,3989],{"className":3933},[611],[569,3935],{"className":3936,"style":3937},[615],"height:1.0019em;vertical-align:-0.25em;",[569,3939,1101],{"className":3940},[730],[569,3942,590],{"className":3943},[620],[569,3945],{"className":3946,"style":625},[624],[569,3948,597],{"className":3949},[629,630],[569,3951,3953,3956],{"className":3952},[813],[569,3954,1115],{"className":3955},[813],[569,3957,3959],{"className":3958},[958],[569,3960,3962],{"className":3961},[738],[569,3963,3965],{"className":3964},[743],[569,3966,3969],{"className":3967,"style":3968},[747],"height:0.7519em;",[569,3970,3971,3974],{"style":970},[569,3972],{"className":3973,"style":974},[755],[569,3975,3977],{"className":3976},[760,761,762,763],[569,3978,3980],{"className":3979},[629,763],[569,3981,1119],{"className":3982},[629,763],[569,3984],{"className":3985,"style":672},[624],[569,3987,1064],{"className":3988},[676],[569,3990],{"className":3991,"style":672},[624],[569,3993,3995,3998],{"className":3994},[611],[569,3996],{"className":3997,"style":723},[615],[569,3999,4001,4004,4075],{"className":4000},[629],[569,4002],{"className":4003},[730,731],[569,4005,4007],{"className":4006},[704],[569,4008,4010,4067],{"className":4009},[738,739],[569,4011,4013,4064],{"className":4012},[743],[569,4014,4016,4042,4050],{"className":4015,"style":748},[747],[569,4017,4018,4021],{"style":751},[569,4019],{"className":4020,"style":756},[755],[569,4022,4024],{"className":4023},[760,761,762,763],[569,4025,4027,4036,4039],{"className":4026},[629,763],[569,4028,4030,4033],{"className":4029},[620,763],[569,4031,1641],{"className":4032},[763],[569,4034,935],{"className":4035},[763],[569,4037],{"className":4038,"style":1634},[624,763],[569,4040,597],{"className":4041},[629,630,763],[569,4043,4044,4047],{"style":772},[569,4045],{"className":4046,"style":756},[755],[569,4048],{"className":4049,"style":780},[779],[569,4051,4052,4055],{"style":783},[569,4053],{"className":4054,"style":756},[755],[569,4056,4058],{"className":4057},[760,761,762,763],[569,4059,4061],{"className":4060},[629,763],[569,4062,708],{"className":4063},[629,763],[569,4065,800],{"className":4066},[799],[569,4068,4070],{"className":4069},[743],[569,4071,4073],{"className":4072,"style":807},[747],[569,4074],{},[569,4076],{"className":4077},[813,731]," schreiben. Im Nenner steht ",[569,4080,4082,4095],{"className":4081},[572],[569,4083,4085],{"className":4084},[576],[578,4086,4087],{"xmlns":580},[582,4088,4089,4093],{},[585,4090,4091],{},[588,4092,597],{},[599,4094,597],{"encoding":601},[569,4096,4098],{"className":4097,"ariaHidden":607},[606],[569,4099,4101,4104],{"className":4100},[611],[569,4102],{"className":4103,"style":1831},[615],[569,4105,597],{"className":4106},[629,630],", nicht ",[569,4109,4111,4128],{"className":4110},[572],[569,4112,4114],{"className":4113},[576],[578,4115,4116],{"xmlns":580},[582,4117,4118,4126],{},[585,4119,4120,4122,4124],{},[588,4121,590],{},[592,4123,594],{},[588,4125,597],{},[599,4127,602],{"encoding":601},[569,4129,4131],{"className":4130,"ariaHidden":607},[606],[569,4132,4134,4137,4140,4143],{"className":4133},[611],[569,4135],{"className":4136,"style":616},[615],[569,4138,590],{"className":4139},[620],[569,4141],{"className":4142,"style":625},[624],[569,4144,597],{"className":4145},[629,630],[3529,4147,3531,4148,4248,4249,4396,4397,4431],{},[569,4149,4151,4181],{"className":4150},[572],[569,4152,4154],{"className":4153},[576],[578,4155,4156],{"xmlns":580},[582,4157,4158,4178],{},[585,4159,4160,4162,4164,4166,4172,4174,4176],{},[588,4161,590],{},[592,4163,594],{},[592,4165,1101],{"stretchy":1474},[929,4167,4168,4170],{},[588,4169,597],{},[650,4171,3019],{},[592,4173,2376],{},[650,4175,708],{},[592,4177,1115],{"stretchy":1474},[599,4179,4180],{"encoding":601},"\\ln(x^2 + 1)",[569,4182,4184,4236],{"className":4183,"ariaHidden":607},[606],[569,4185,4187,4191,4194,4197,4227,4230,4233],{"className":4186},[611],[569,4188],{"className":4189,"style":4190},[615],"height:1.0641em;vertical-align:-0.25em;",[569,4192,590],{"className":4193},[620],[569,4195,1101],{"className":4196},[730],[569,4198,4200,4203],{"className":4199},[629],[569,4201,597],{"className":4202},[629,630],[569,4204,4206],{"className":4205},[958],[569,4207,4209],{"className":4208},[738],[569,4210,4212],{"className":4211},[743],[569,4213,4216],{"className":4214,"style":4215},[747],"height:0.8141em;",[569,4217,4218,4221],{"style":970},[569,4219],{"className":4220,"style":974},[755],[569,4222,4224],{"className":4223},[760,761,762,763],[569,4225,3019],{"className":4226},[629,763],[569,4228],{"className":4229,"style":1993},[624],[569,4231,2376],{"className":4232},[1997],[569,4234],{"className":4235,"style":1993},[624],[569,4237,4239,4242,4245],{"className":4238},[611],[569,4240],{"className":4241,"style":2395},[615],[569,4243,708],{"className":4244},[629],[569,4246,1115],{"className":4247},[813]," nur ",[569,4250,4252,4280],{"className":4251},[572],[569,4253,4255],{"className":4254},[576],[578,4256,4257],{"xmlns":580},[582,4258,4259,4277],{},[585,4260,4261],{},[704,4262,4263,4265],{},[650,4264,708],{},[585,4266,4267,4273,4275],{},[929,4268,4269,4271],{},[588,4270,597],{},[650,4272,3019],{},[592,4274,2376],{},[650,4276,708],{},[599,4278,4279],{"encoding":601},"\\frac{1}{x^2 + 1}",[569,4281,4283],{"className":4282,"ariaHidden":607},[606],[569,4284,4286,4290],{"className":4285},[611],[569,4287],{"className":4288,"style":4289},[615],"height:1.2484em;vertical-align:-0.4033em;",[569,4291,4293,4296,4393],{"className":4292},[629],[569,4294],{"className":4295},[730,731],[569,4297,4299],{"className":4298},[704],[569,4300,4302,4384],{"className":4301},[738,739],[569,4303,4305,4381],{"className":4304},[743],[569,4306,4308,4359,4367],{"className":4307,"style":748},[747],[569,4309,4310,4313],{"style":751},[569,4311],{"className":4312,"style":756},[755],[569,4314,4316],{"className":4315},[760,761,762,763],[569,4317,4319,4353,4356],{"className":4318},[629,763],[569,4320,4322,4325],{"className":4321},[629,763],[569,4323,597],{"className":4324},[629,630,763],[569,4326,4328],{"className":4327},[958],[569,4329,4331],{"className":4330},[738],[569,4332,4334],{"className":4333},[743],[569,4335,4338],{"className":4336,"style":4337},[747],"height:0.7463em;",[569,4339,4341,4345],{"style":4340},"top:-2.786em;margin-right:0.0714em;",[569,4342],{"className":4343,"style":4344},[755],"height:2.5em;",[569,4346,4350],{"className":4347},[760,4348,4349,763],"reset-size3","size1",[569,4351,3019],{"className":4352},[629,763],[569,4354,2376],{"className":4355},[1997,763],[569,4357,708],{"className":4358},[629,763],[569,4360,4361,4364],{"style":772},[569,4362],{"className":4363,"style":756},[755],[569,4365],{"className":4366,"style":780},[779],[569,4368,4369,4372],{"style":783},[569,4370],{"className":4371,"style":756},[755],[569,4373,4375],{"className":4374},[760,761,762,763],[569,4376,4378],{"className":4377},[629,763],[569,4379,708],{"className":4380},[629,763],[569,4382,800],{"className":4383},[799],[569,4385,4387],{"className":4386},[743],[569,4388,4391],{"className":4389,"style":4390},[747],"height:0.4033em;",[569,4392],{},[569,4394],{"className":4395},[813,731]," angeben. Die innere Ableitung ",[569,4398,4400,4416],{"className":4399},[572],[569,4401,4403],{"className":4402},[576],[578,4404,4405],{"xmlns":580},[582,4406,4407,4413],{},[585,4408,4409,4411],{},[650,4410,3019],{},[588,4412,597],{},[599,4414,4415],{"encoding":601},"2x",[569,4417,4419],{"className":4418,"ariaHidden":607},[606],[569,4420,4422,4425,4428],{"className":4421},[611],[569,4423],{"className":4424,"style":686},[615],[569,4426,3019],{"className":4427},[629],[569,4429,597],{"className":4430},[629,630]," gehört in den Zähler.",{"title":4433,"searchDepth":4434,"depth":4434,"links":4435},"",2,[4436,4437,4438,4439],{"id":562,"depth":4434,"text":563},{"id":1694,"depth":4434,"text":1695},{"id":2336,"depth":4434,"text":2337},{"id":3523,"depth":4434,"text":3524},[4441,4443,4445,4449,4452,4455,4457,4459],{"lhs":4442,"rhs":652},"round((log(x+1e-5) - log(x-1e-5))\u002F(2e-5) - 1\u002Fx, 6)",{"lhs":4444,"rhs":597},"exp(log(x))",{"lhs":4446,"rhs":652,"where":4447},"round((log(abs(x+1e-5)) - log(abs(x-1e-5)))\u002F(2e-5) - 1\u002Fx, 6)",{"x":4448},-2,{"lhs":4450,"rhs":4451},"log(3*x)","log(3) + log(x)",{"lhs":4453,"rhs":4454},"3\u002F(3*x)","1\u002Fx",{"lhs":4456,"rhs":652},"round((log((x+1e-5)^2 + 1) - log((x-1e-5)^2 + 1))\u002F(2e-5) - 2*x\u002F(x^2 + 1), 6)",{"lhs":4458,"rhs":708},"2*1\u002F(1^2 + 1)",{"lhs":4460,"rhs":652},"round((log(x+1e-5, 10) - log(x-1e-5, 10))\u002F(2e-5) - 1\u002F(x*log(10)), 6)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x > 0\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\">x\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:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","derivative-ln","f(x) = \\ln x \\;\\Rightarrow\\; f'(x) = \\frac{1}{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>ln\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\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>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f(x) = \\ln x \\;\\Rightarrow\\; f'(x) = \\frac{1}{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\" 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