[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.53,0:00:05.96,Default,,0000,0000,0000,,-可微函数x和y由以下方程关联
Dialogue: 0,0:00:05.96,0:00:08.72,Default,,0000,0000,0000,,Sin(x)+Cos(y)
Dialogue: 0,0:00:09.58,0:00:12.48,Default,,0000,0000,0000,,将等于2的平方根
Dialogue: 0,0:00:12.48,0:00:17.90,Default,,0000,0000,0000,,它也告诉我们dx/dt=5
Dialogue: 0,0:00:17.90,0:00:25.14,Default,,0000,0000,0000,,他也告诉我们要在当y=π/4且x大于0小于π/2时
Dialogue: 0,0:00:25.14,0:00:29.79,Default,,0000,0000,0000,,求出基于t的y的导数
Dialogue: 0,0:00:29.79,0:00:32.36,Default,,0000,0000,0000,,所以他告诉了我们基于t的x的导数
Dialogue: 0,0:00:32.36,0:00:34.68,Default,,0000,0000,0000,,并且我们要找到
Dialogue: 0,0:00:34.68,0:00:37.13,Default,,0000,0000,0000,,基于t的y的导数
Dialogue: 0,0:00:37.13,0:00:41.30,Default,,0000,0000,0000,,认为x和y都是基于t的函数的假设是合理的
Dialogue: 0,0:00:42.40,0:00:45.75,Default,,0000,0000,0000,,所以你甚至可以在这将函数重新列一遍
Dialogue: 0,0:00:45.75,0:00:48.58,Default,,0000,0000,0000,,你可以将其重新写为sin(x)，其中
Dialogue: 0,0:00:50.68,0:00:53.04,Default,,0000,0000,0000,,x是t的函数
Dialogue: 0,0:00:53.04,0:00:53.96,Default,,0000,0000,0000,,加上
Dialogue: 0,0:00:55.82,0:00:58.40,Default,,0000,0000,0000,,cos(y)，y也是t的函数
Dialogue: 0,0:00:59.50,0:01:02.08,Default,,0000,0000,0000,,整个式子等于2的平方根
Dialogue: 0,0:01:02.92,0:01:04.54,Default,,0000,0000,0000,,现在你可能有些困惑
Dialogue: 0,0:01:04.54,0:01:06.38,Default,,0000,0000,0000,,你未曾将x设为具有3个未知量的函数
Dialogue: 0,0:01:06.38,0:01:10.07,Default,,0000,0000,0000,,或你未曾将y当做一个除了x外另有变量的函数
Dialogue: 0,0:01:10.07,0:01:11.74,Default,,0000,0000,0000,,但请记住，x和y仅是自变量
Dialogue: 0,0:01:11.74,0:01:15.42,Default,,0000,0000,0000,,这可以是f(t),那可以是g(t)
Dialogue: 0,0:01:15.42,0:01:17.60,Default,,0000,0000,0000,,而不是x(t)或y(t)
Dialogue: 0,0:01:17.60,0:01:19.50,Default,,0000,0000,0000,,这也许会让你觉得自然些
Dialogue: 0,0:01:19.50,0:01:23.17,Default,,0000,0000,0000,,不必说如果我们要求出dt
Dialogue: 0,0:01:24.19,0:01:29.95,Default,,0000,0000,0000,,我们要做的是对这个方程两边的 t 求导
Dialogue: 0,0:01:29.95,0:01:31.36,Default,,0000,0000,0000,,所以让我们处理它吧
Dialogue: 0,0:01:31.36,0:01:33.30,Default,,0000,0000,0000,,所以我们将从等式左侧入手
Dialogue: 0,0:01:33.30,0:01:38.04,Default,,0000,0000,0000,,所以我们将把它与 t 相关联，对 t 取它的导数。
Dialogue: 0,0:01:38.04,0:01:41.00,Default,,0000,0000,0000,,我们将对t取它的导数
Dialogue: 0,0:01:41.00,0:01:42.34,Default,,0000,0000,0000,,接着我们将取等式右侧这一常数项
Dialogue: 0,0:01:42.34,0:01:46.55,Default,,0000,0000,0000,,对t的导数
Dialogue: 0,0:01:46.55,0:01:49.76,Default,,0000,0000,0000,,所以我们逐一思考这些式子
Dialogue: 0,0:01:49.76,0:01:51.44,Default,,0000,0000,0000,,所以这是什么呢
Dialogue: 0,0:01:51.44,0:01:53.11,Default,,0000,0000,0000,,让我换种颜色
Dialogue: 0,0:01:53.11,0:01:56.62,Default,,0000,0000,0000,,我正在用水笔解的式子
Dialogue: 0,0:01:56.62,0:01:58.24,Default,,0000,0000,0000,,我该如何写它呢
Dialogue: 0,0:01:58.24,0:02:00.40,Default,,0000,0000,0000,,所以我取对t的函数
Dialogue: 0,0:02:00.40,0:02:04.92,Default,,0000,0000,0000,,我有某数的sin值，其自身为关于t的函数
Dialogue: 0,0:02:04.92,0:02:07.77,Default,,0000,0000,0000,,所以我将在这里应用链式法则
Dialogue: 0,0:02:07.77,0:02:14.65,Default,,0000,0000,0000,,首先，我将求sinx的导数
Dialogue: 0,0:02:16.51,0:02:18.71,Default,,0000,0000,0000,,我将之写作sinx(t))
Dialogue: 0,0:02:18.71,0:02:22.36,Default,,0000,0000,0000,,但为了简化，我将其恢复为原式
Dialogue: 0,0:02:22.36,0:02:25.24,Default,,0000,0000,0000,,接着我将其与x的导数相乘
Dialogue: 0,0:02:25.24,0:02:28.77,Default,,0000,0000,0000,,你可以说，对于t的导数
Dialogue: 0,0:02:28.77,0:02:32.78,Default,,0000,0000,0000,,乘上dx/dt
Dialogue: 0,0:02:32.78,0:02:34.51,Default,,0000,0000,0000,,这和你之前处理链式法则相比，
Dialogue: 0,0:02:34.51,0:02:35.59,Default,,0000,0000,0000,,可能有些反常
Dialogue: 0,0:02:35.59,0:02:38.74,Default,,0000,0000,0000,,之前我们只处理与x或y相关的
Dialogue: 0,0:02:38.74,0:02:41.27,Default,,0000,0000,0000,,但这就是现在发生的，我将取sin外的
Dialogue: 0,0:02:41.27,0:02:43.56,Default,,0000,0000,0000,,对于某值的某值的导数，
Dialogue: 0,0:02:43.56,0:02:46.55,Default,,0000,0000,0000,,在这种情况下，即为x
Dialogue: 0,0:02:46.55,0:02:48.50,Default,,0000,0000,0000,,接着我将求某值的导数
Dialogue: 0,0:02:48.50,0:02:51.42,Default,,0000,0000,0000,,在这种情况下，即为x对于t的导数
Dialogue: 0,0:02:51.42,0:02:53.93,Default,,0000,0000,0000,,我们可以对第二个多项式
Dialogue: 0,0:02:53.93,0:02:56.01,Default,,0000,0000,0000,,用同样的方式处理
Dialogue: 0,0:02:56.99,0:03:01.22,Default,,0000,0000,0000,,所以我将求d/dy乘上，
Dialogue: 0,0:03:01.22,0:03:04.33,Default,,0000,0000,0000,,我猜你将说
Dialogue: 0,0:03:04.33,0:03:05.58,Default,,0000,0000,0000,,cos(y)
Dialogue: 0,0:03:07.69,0:03:09.21,Default,,0000,0000,0000,,接着我将之相乘：
Dialogue: 0,0:03:09.21,0:03:12.87,Default,,0000,0000,0000,,乘上对于t的y的导数
Dialogue: 0,0:03:14.26,0:03:17.45,Default,,0000,0000,0000,,接着它们整体将等于多少呢
Dialogue: 0,0:03:17.45,0:03:20.74,Default,,0000,0000,0000,,对于t的常数的导数
Dialogue: 0,0:03:20.74,0:03:22.16,Default,,0000,0000,0000,,根号2是个常数
Dialogue: 0,0:03:22.16,0:03:23.91,Default,,0000,0000,0000,,将不会随t的变化而变化
Dialogue: 0,0:03:23.91,0:03:27.38,Default,,0000,0000,0000,,所以它的导数，其变化率即为0
Dialogue: 0,0:03:27.38,0:03:29.63,Default,,0000,0000,0000,,好的，现在我们求到了
Dialogue: 0,0:03:29.63,0:03:31.36,Default,,0000,0000,0000,,所有的值
Dialogue: 0,0:03:31.36,0:03:33.68,Default,,0000,0000,0000,,所以第一步，sin(x)的导数为
Dialogue: 0,0:03:33.68,0:03:38.28,Default,,0000,0000,0000,,cos(x)乘x对于t的导数
Dialogue: 0,0:03:38.28,0:03:40.27,Default,,0000,0000,0000,,我将在这写下
Dialogue: 0,0:03:40.27,0:03:42.21,Default,,0000,0000,0000,,x对于t导数
Dialogue: 0,0:03:42.21,0:03:44.96,Default,,0000,0000,0000,,接着我们将有，这里是加上
Dialogue: 0,0:03:44.96,0:03:47.16,Default,,0000,0000,0000,,y对于t的导数
Dialogue: 0,0:03:47.16,0:03:51.01,Default,,0000,0000,0000,,所以加上y对于t的导数
Dialogue: 0,0:03:51.01,0:03:52.44,Default,,0000,0000,0000,,我在这把顺序调换了下
Dialogue: 0,0:03:52.44,0:03:54.47,Default,,0000,0000,0000,,所以它将提前
Dialogue: 0,0:03:54.47,0:03:58.37,Default,,0000,0000,0000,,现在，对于y的cos(y)的导数是多少呢？
Dialogue: 0,0:03:58.37,0:04:01.34,Default,,0000,0000,0000,,其值为-sin(y)
Dialogue: 0,0:04:01.34,0:04:05.26,Default,,0000,0000,0000,,接着，让我先把sin(y)写在这
Dialogue: 0,0:04:05.26,0:04:07.12,Default,,0000,0000,0000,,再加个负号
Dialogue: 0,0:04:07.12,0:04:10.12,Default,,0000,0000,0000,,将这（符号）擦除并换为减号
Dialogue: 0,0:04:11.60,0:04:15.10,Default,,0000,0000,0000,,而这依旧将等于0
Dialogue: 0,0:04:16.06,0:04:18.74,Default,,0000,0000,0000,,因此我们可以求出什么呢
Dialogue: 0,0:04:18.74,0:04:21.78,Default,,0000,0000,0000,,它告诉我们x对于t的导数的值为5
Dialogue: 0,0:04:21.78,0:04:25.40,Default,,0000,0000,0000,,他就在这告诉了我们
Dialogue: 0,0:04:25.40,0:04:27.48,Default,,0000,0000,0000,,所以这将等于5
Dialogue: 0,0:04:29.09,0:04:32.68,Default,,0000,0000,0000,,我们想求出对于t的y的导数的值
Dialogue: 0,0:04:32.68,0:04:36.14,Default,,0000,0000,0000,,他告诉了我们y的值，π/4
Dialogue: 0,0:04:36.14,0:04:40.31,Default,,0000,0000,0000,,这里，y是π/4，所以写下π/4
Dialogue: 0,0:04:41.61,0:04:43.79,Default,,0000,0000,0000,,所以我们看一下，我们得求出它
Dialogue: 0,0:04:43.79,0:04:45.72,Default,,0000,0000,0000,,我们仍有两个未知量
Dialogue: 0,0:04:45.72,0:04:47.45,Default,,0000,0000,0000,,我们不知道x的值，我们也不知道
Dialogue: 0,0:04:47.45,0:04:49.58,Default,,0000,0000,0000,,y对于t的导数值
Dialogue: 0,0:04:49.58,0:04:51.10,Default,,0000,0000,0000,,那就是我们所需求的值
Dialogue: 0,0:04:51.10,0:04:52.47,Default,,0000,0000,0000,,所以x的值是多少呢？
Dialogue: 0,0:04:52.47,0:04:55.42,Default,,0000,0000,0000,,当y等于π/4时，x的值是多少呢
Dialogue: 0,0:04:55.42,0:04:56.44,Default,,0000,0000,0000,,好的，为了求出它，
Dialogue: 0,0:04:56.44,0:05:00.22,Default,,0000,0000,0000,,我们将回到原式
Dialogue: 0,0:05:00.22,0:05:03.75,Default,,0000,0000,0000,,所以当y等于π/4时，你会得到
Dialogue: 0,0:05:03.75,0:05:04.85,Default,,0000,0000,0000,,让我把它写下
Dialogue: 0,0:05:04.85,0:05:05.68,Default,,0000,0000,0000,,sin(x)，加上
Dialogue: 0,0:05:07.39,0:05:08.81,Default,,0000,0000,0000,,cos(π/4),
Dialogue: 0,0:05:10.72,0:05:13.98,Default,,0000,0000,0000,,（cos(π/4)）的值为二分之根号二
Dialogue: 0,0:05:13.98,0:05:15.90,Default,,0000,0000,0000,,cos(π/4)
Dialogue: 0,0:05:17.59,0:05:20.94,Default,,0000,0000,0000,,我们可以转化为我们（熟悉的角度）或用单位圆解决
Dialogue: 0,0:05:20.94,0:05:22.56,Default,,0000,0000,0000,,我们正处于第一象限中，
Dialogue: 0,0:05:22.56,0:05:24.16,Default,,0000,0000,0000,,如果我们以角度制思考，它的值为45°
Dialogue: 0,0:05:24.16,0:05:28.07,Default,,0000,0000,0000,,它将等于二分之根号二
Dialogue: 0,0:05:28.07,0:05:30.58,Default,,0000,0000,0000,,我们可从等式两侧同时减去二分之根号二，
Dialogue: 0,0:05:30.58,0:05:32.85,Default,,0000,0000,0000,,我们将的值
Dialogue: 0,0:05:32.85,0:05:37.71,Default,,0000,0000,0000,,sin(x)的值为，好的，如果你从根号二中减去
Dialogue: 0,0:05:37.71,0:05:39.47,Default,,0000,0000,0000,,二分之根号二，
Dialogue: 0,0:05:39.47,0:05:40.76,Default,,0000,0000,0000,,你将减去它的一半
Dialogue: 0,0:05:40.76,0:05:42.22,Default,,0000,0000,0000,,所以你仍留有它的一半
Dialogue: 0,0:05:42.22,0:05:44.71,Default,,0000,0000,0000,,所以就是二分之根号二
Dialogue: 0,0:05:44.71,0:05:48.72,Default,,0000,0000,0000,,所以，x的值是多少呢，我将取它的sin值
Dialogue: 0,0:05:48.72,0:05:50.77,Default,,0000,0000,0000,,记住，这是在角度制中
Dialogue: 0,0:05:50.77,0:05:52.36,Default,,0000,0000,0000,,如果我们用单位圆思考，它将在第一象限。
Dialogue: 0,0:05:52.36,0:05:54.78,Default,,0000,0000,0000,,在这情况下x将是一个角，
Dialogue: 0,0:05:54.78,0:05:56.08,Default,,0000,0000,0000,,就在这
Dialogue: 0,0:05:56.08,0:05:59.38,Default,,0000,0000,0000,,所以那将是又一次的π/4
Dialogue: 0,0:05:59.38,0:06:03.16,Default,,0000,0000,0000,,所以x的值为π/4
Dialogue: 0,0:06:03.16,0:06:05.83,Default,,0000,0000,0000,,当y等于π/4
Dialogue: 0,0:06:05.83,0:06:09.48,Default,,0000,0000,0000,,
Dialogue: 0,0:06:09.48,0:06:11.44,Default,,0000,0000,0000,,
Dialogue: 0,0:06:11.44,0:06:13.46,Default,,0000,0000,0000,,
Dialogue: 0,0:06:13.46,0:06:15.63,Default,,0000,0000,0000,,
Dialogue: 0,0:06:17.52,0:06:19.35,Default,,0000,0000,0000,,
Dialogue: 0,0:06:22.21,0:06:23.05,Default,,0000,0000,0000,,
Dialogue: 0,0:06:24.27,0:06:26.97,Default,,0000,0000,0000,,
Dialogue: 0,0:06:26.97,0:06:28.77,Default,,0000,0000,0000,,
Dialogue: 0,0:06:28.77,0:06:31.02,Default,,0000,0000,0000,,
Dialogue: 0,0:06:32.56,0:06:33.98,Default,,0000,0000,0000,,
Dialogue: 0,0:06:35.40,0:06:38.52,Default,,0000,0000,0000,,
Dialogue: 0,0:06:38.52,0:06:40.71,Default,,0000,0000,0000,,
Dialogue: 0,0:06:40.71,0:06:43.45,Default,,0000,0000,0000,,
Dialogue: 0,0:06:43.45,0:06:45.11,Default,,0000,0000,0000,,
Dialogue: 0,0:06:45.11,0:06:46.93,Default,,0000,0000,0000,,
Dialogue: 0,0:06:46.93,0:06:49.68,Default,,0000,0000,0000,,
Dialogue: 0,0:06:49.68,0:06:53.84,Default,,0000,0000,0000,,
Dialogue: 0,0:06:54.75,0:06:57.58,Default,,0000,0000,0000,,
Dialogue: 0,0:06:57.58,0:07:00.88,Default,,0000,0000,0000,,
Dialogue: 0,0:07:00.88,0:07:02.24,Default,,0000,0000,0000,,
Dialogue: 0,0:07:02.24,0:07:04.55,Default,,0000,0000,0000,,
Dialogue: 0,0:07:04.55,0:07:05.88,Default,,0000,0000,0000,,
Dialogue: 0,0:07:05.88,0:07:08.18,Default,,0000,0000,0000,,
Dialogue: 0,0:07:08.18,0:07:10.04,Default,,0000,0000,0000,,
Dialogue: 0,0:07:10.04,0:07:11.28,Default,,0000,0000,0000,,
Dialogue: 0,0:07:11.28,0:07:13.23,Default,,0000,0000,0000,,
Dialogue: 0,0:07:13.23,0:07:15.50,Default,,0000,0000,0000,,
Dialogue: 0,0:07:15.50,0:07:17.75,Default,,0000,0000,0000,,
Dialogue: 0,0:07:17.75,0:07:19.84,Default,,0000,0000,0000,,
Dialogue: 0,0:07:19.84,0:07:23.07,Default,,0000,0000,0000,,
Dialogue: 0,0:07:23.07,0:07:26.63,Default,,0000,0000,0000,,
Dialogue: 0,0:07:26.63,0:07:28.04,Default,,0000,0000,0000,,
Dialogue: 0,0:07:29.56,0:07:30.73,Default,,0000,0000,0000,,
Dialogue: 0,0:07:30.73,0:07:33.65,Default,,0000,0000,0000,,
Dialogue: 0,0:07:33.65,0:07:37.82,Default,,0000,0000,0000,,
Dialogue: 0,0:07:38.68,0:07:42.29,Default,,0000,0000,0000,,
Dialogue: 0,0:07:42.29,0:07:44.66,Default,,0000,0000,0000,,
Dialogue: 0,0:07:44.66,0:07:47.100,Default,,0000,0000,0000,,
Dialogue: 0,0:07:47.100,0:07:50.17,Default,,0000,0000,0000,,