[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.84,Default,,0000,0000,0000,,有人要求讲一下
Dialogue: 0,0:00:00.84,0:00:04.09,Default,,0000,0000,0000,,证明√x的导数
Dialogue: 0,0:00:04.09,0:00:06.30,Default,,0000,0000,0000,,所以这集快速证明一下
Dialogue: 0,0:00:06.30,0:00:08.30,Default,,0000,0000,0000,,√x的导数
Dialogue: 0,0:00:08.30,0:00:10.37,Default,,0000,0000,0000,,由导数的定义可知
Dialogue: 0,0:00:10.37,0:00:13.68,Default,,0000,0000,0000,,函数√x的导数
Dialogue: 0,0:00:13.68,0:00:22.28,Default,,0000,0000,0000,,换颜色 更多样一些
Dialogue: 0,0:00:22.28,0:00:26.52,Default,,0000,0000,0000,,它等于Δx趋于0时的极限
Dialogue: 0,0:00:26.52,0:00:33.08,Default,,0000,0000,0000,,有人也说h趋于0
Dialogue: 0,0:00:33.08,0:00:35.60,Default,,0000,0000,0000,,或d趋于0 这里用Δx
Dialogue: 0,0:00:35.60,0:00:36.36,Default,,0000,0000,0000,,Δx趋于0时
Dialogue: 0,0:00:36.36,0:00:37.45,Default,,0000,0000,0000,,然后f(x+Δx)
Dialogue: 0,0:00:37.45,0:00:39.45,Default,,0000,0000,0000,,此处这是f(x)
Dialogue: 0,0:00:39.45,0:00:41.83,Default,,0000,0000,0000,,所以它是√(x+Δx)-f(x)
Dialogue: 0,0:00:41.83,0:00:42.91,Default,,0000,0000,0000,,此处即√x
Dialogue: 0,0:00:42.91,0:00:52.26,Default,,0000,0000,0000,,整体/x的变化量Δx
Dialogue: 0,0:00:52.26,0:00:54.64,Default,,0000,0000,0000,,现在看这个式子
Dialogue: 0,0:00:54.64,0:00:57.14,Default,,0000,0000,0000,,还不能化简
Dialogue: 0,0:00:57.14,0:01:00.04,Default,,0000,0000,0000,,使它变得适于分析
Dialogue: 0,0:01:00.04,0:01:02.58,Default,,0000,0000,0000,,那么分子和分母同乘以
Dialogue: 0,0:01:02.58,0:01:04.94,Default,,0000,0000,0000,,分子的共轭根式 什么意思呢？
Dialogue: 0,0:01:04.94,0:01:09.94,Default,,0000,0000,0000,,把它写下来
Dialogue: 0,0:01:09.94,0:01:12.54,Default,,0000,0000,0000,,Δx趋于0时的极限
Dialogue: 0,0:01:12.54,0:01:13.79,Default,,0000,0000,0000,,把这再写下来
Dialogue: 0,0:01:13.79,0:01:14.20,Default,,0000,0000,0000,,即√(x+Δx)
Dialogue: 0,0:01:14.20,0:01:15.48,Default,,0000,0000,0000,,-√x
Dialogue: 0,0:01:15.48,0:01:19.74,Default,,0000,0000,0000,,整体/Δx
Dialogue: 0,0:01:19.74,0:01:21.28,Default,,0000,0000,0000,,换颜色 将它乘以
Dialogue: 0,0:01:21.28,0:01:26.65,Default,,0000,0000,0000,,√(x+Δx)
Dialogue: 0,0:01:26.65,0:01:28.61,Default,,0000,0000,0000,,+√x 除以
Dialogue: 0,0:01:28.61,0:01:31.20,Default,,0000,0000,0000,,√(x+Δx)+√x
Dialogue: 0,0:01:31.20,0:01:34.49,Default,,0000,0000,0000,,这只是1 显然可以相乘
Dialogue: 0,0:01:34.49,0:01:41.84,Default,,0000,0000,0000,,假设x和Δx都不为0
Dialogue: 0,0:01:41.84,0:01:48.26,Default,,0000,0000,0000,,这是一个定值 为1
Dialogue: 0,0:01:48.26,0:01:49.25,Default,,0000,0000,0000,,可以相乘
Dialogue: 0,0:01:49.25,0:01:53.42,Default,,0000,0000,0000,,这是1/1 将它乘以这个式子
Dialogue: 0,0:01:53.42,0:01:57.11,Default,,0000,0000,0000,,得到当Δx趋于0时
Dialogue: 0,0:01:57.11,0:01:59.09,Default,,0000,0000,0000,,这是(a-b)(a+b)
Dialogue: 0,0:01:59.09,0:02:00.01,Default,,0000,0000,0000,,在这旁边列出来
Dialogue: 0,0:02:00.01,0:02:02.13,Default,,0000,0000,0000,,(a+b)(a-b)
Dialogue: 0,0:02:02.13,0:02:10.90,Default,,0000,0000,0000,,=a^2-b^2
Dialogue: 0,0:02:10.90,0:02:13.51,Default,,0000,0000,0000,,这是(a+b)(a-b)
Dialogue: 0,0:02:13.51,0:02:15.36,Default,,0000,0000,0000,,所以它等于a^2
Dialogue: 0,0:02:15.36,0:02:20.88,Default,,0000,0000,0000,,那么这个量的平方是什么
Dialogue: 0,0:02:20.88,0:02:23.15,Default,,0000,0000,0000,,两个都可以 都代表a
Dialogue: 0,0:02:23.15,0:02:26.60,Default,,0000,0000,0000,,是x+Δx
Dialogue: 0,0:02:26.60,0:02:29.41,Default,,0000,0000,0000,,那么得到x+Δx
Dialogue: 0,0:02:29.41,0:02:32.01,Default,,0000,0000,0000,,然后b^2呢？
Dialogue: 0,0:02:32.01,0:02:33.18,Default,,0000,0000,0000,,-√x类比为b
Dialogue: 0,0:02:33.18,0:02:35.45,Default,,0000,0000,0000,,那么(√x)^2=x
Dialogue: 0,0:02:35.45,0:02:39.43,Default,,0000,0000,0000,,整体/(Δx・
Dialogue: 0,0:02:39.43,0:02:41.05,Default,,0000,0000,0000,,(√(x+Δx)+√x))
Dialogue: 0,0:02:41.05,0:02:46.38,Default,,0000,0000,0000,,看看怎么化简
Dialogue: 0,0:02:46.38,0:02:50.64,Default,,0000,0000,0000,,这里有x和-x
Dialogue: 0,0:02:50.64,0:02:56.76,Default,,0000,0000,0000,,两者抵消了 x-x
Dialogue: 0,0:02:56.76,0:03:04.21,Default,,0000,0000,0000,,然后分子和分母上剩下的
Dialogue: 0,0:03:04.21,0:03:05.90,Default,,0000,0000,0000,,都有Δx
Dialogue: 0,0:03:05.90,0:03:08.58,Default,,0000,0000,0000,,分子和分母同除以Δx
Dialogue: 0,0:03:08.58,0:03:11.48,Default,,0000,0000,0000,,都变为1
Dialogue: 0,0:03:11.48,0:03:13.46,Default,,0000,0000,0000,,所以它= 写的小点
Dialogue: 0,0:03:13.46,0:03:15.69,Default,,0000,0000,0000,,因为没地方了
Dialogue: 0,0:03:15.69,0:03:18.77,Default,,0000,0000,0000,,=Δx趋于0时的极限 1/
Dialogue: 0,0:03:18.77,0:03:22.82,Default,,0000,0000,0000,,显然只有假设Δx--
Dialogue: 0,0:03:22.82,0:03:26.35,Default,,0000,0000,0000,,一开始除以Δx
Dialogue: 0,0:03:26.35,0:03:34.92,Default,,0000,0000,0000,,所以它不为0 只是趋于0
Dialogue: 0,0:03:34.92,0:03:37.78,Default,,0000,0000,0000,,那么得到√(x+Δx)
Dialogue: 0,0:03:37.78,0:03:40.22,Default,,0000,0000,0000,,+√x
Dialogue: 0,0:03:40.22,0:03:42.42,Default,,0000,0000,0000,,现在直接求Δx趋于0时
Dialogue: 0,0:03:42.42,0:03:50.32,Default,,0000,0000,0000,,它的极限
Dialogue: 0,0:03:50.32,0:03:51.86,Default,,0000,0000,0000,,只需设Δx=0
Dialogue: 0,0:03:51.86,0:03:53.55,Default,,0000,0000,0000,,即趋于0的含义
Dialogue: 0,0:03:53.55,0:03:54.41,Default,,0000,0000,0000,,那么等于1/√x
Dialogue: 0,0:03:54.41,0:03:56.44,Default,,0000,0000,0000,,对吗？Δx=0 可以忽略
Dialogue: 0,0:03:56.44,0:03:58.14,Default,,0000,0000,0000,,取其极限一直到0
Dialogue: 0,0:03:58.14,0:04:04.26,Default,,0000,0000,0000,,然后这显然还是√x
Dialogue: 0,0:04:04.26,0:04:06.79,Default,,0000,0000,0000,,+√x
Dialogue: 0,0:04:06.79,0:04:09.12,Default,,0000,0000,0000,,=1/(2√x)
Dialogue: 0,0:04:09.12,0:04:13.00,Default,,0000,0000,0000,,=(1/2)x^(-1/2)
Dialogue: 0,0:04:13.00,0:04:17.16,Default,,0000,0000,0000,,刚才证明了x^(1/2)
Dialogue: 0,0:04:17.16,0:04:19.35,Default,,0000,0000,0000,,的导数为(1/2)x^(-1/2)
Dialogue: 0,0:04:19.35,0:04:24.89,Default,,0000,0000,0000,,它跟一般情况一致
Dialogue: 0,0:04:24.89,0:04:28.90,Default,,0000,0000,0000,,即求导
Dialogue: 0,0:04:28.90,0:04:35.22,Default,,0000,0000,0000,,x^n的导数为
Dialogue: 0,0:04:35.22,0:04:41.70,Default,,0000,0000,0000,,n・x^(n-1) 即使此处n=1/2
Dialogue: 0,0:04:41.70,0:04:50.85,Default,,0000,0000,0000,,也是合乎情况的
Dialogue: 0,0:04:50.85,0:04:55.15,Default,,0000,0000,0000,,未就所有的分数都给出证明 但这是开始
Dialogue: 0,0:04:55.15,0:04:56.10,Default,,0000,0000,0000,,这是常见的一个 √x
Dialogue: 0,0:04:56.10,0:04:58.96,Default,,0000,0000,0000,,好在证明起来不很复杂
Dialogue: 0,0:04:58.96,0:05:01.12,Default,,0000,0000,0000,,下一集再见