1
00:00:00,534 --> 00:00:05,965
-可微函数x和y由以下方程关联

2
00:00:05,965 --> 00:00:08,715
Sin(x)+Cos(y)

3
00:00:09,578 --> 00:00:12,479
将等于2的平方根

4
00:00:12,479 --> 00:00:17,897
它也告诉我们dx/dt=5

5
00:00:17,897 --> 00:00:25,145
他也告诉我们要在当y=π/4且x大于0小于π/2时

6
00:00:25,145 --> 00:00:29,789
求出基于t的y的导数

7
00:00:29,789 --> 00:00:32,365
所以他告诉了我们基于t的x的导数

8
00:00:32,365 --> 00:00:34,685
并且我们要找到

9
00:00:34,685 --> 00:00:37,134
基于t的y的导数

10
00:00:37,134 --> 00:00:41,301
认为x和y都是基于t的函数的假设是合理的

11
00:00:42,403 --> 00:00:45,751
所以你甚至可以在这将函数重新列一遍

12
00:00:45,751 --> 00:00:48,584
你可以将其重新写为sin(x)，其中

13
00:00:50,683 --> 00:00:53,040
x是t的函数

14
00:00:53,040 --> 00:00:53,957
加上

15
00:00:55,815 --> 00:00:58,398
cos(y)，y也是t的函数

16
00:00:59,499 --> 00:01:02,082
整个式子等于2的平方根

17
00:01:02,920 --> 00:01:04,542
现在你可能有些困惑

18
00:01:04,542 --> 00:01:06,376
你未曾将x设为具有3个未知量的函数

19
00:01:06,376 --> 00:01:10,068
或你未曾将y当做一个除了x外另有变量的函数

20
00:01:10,068 --> 00:01:11,740
但请记住，x和y仅是自变量

21
00:01:11,740 --> 00:01:15,419
这可以是f(t),那可以是g(t)

22
00:01:15,419 --> 00:01:17,605
而不是x(t)或y(t)

23
00:01:17,605 --> 00:01:19,503
这也许会让你觉得自然些

24
00:01:19,503 --> 00:01:23,170
不必说如果我们要求出dt

25
00:01:24,186 --> 00:01:29,952
我们要做的是对这个方程两边的 t 求导

26
00:01:29,952 --> 00:01:31,357
所以让我们处理它吧

27
00:01:31,357 --> 00:01:33,303
所以我们将从等式左侧入手

28
00:01:33,303 --> 00:01:38,035
所以我们将把它与 t 相关联，对 t 取它的导数。

29
00:01:38,035 --> 00:01:41,001
我们将对t取它的导数

30
00:01:41,001 --> 00:01:42,338
接着我们将取等式右侧这一常数项

31
00:01:42,338 --> 00:01:46,547
对t的导数

32
00:01:46,547 --> 00:01:49,764
所以我们逐一思考这些式子

33
00:01:49,764 --> 00:01:51,444
所以这是什么呢

34
00:01:51,444 --> 00:01:53,114
让我换种颜色

35
00:01:53,114 --> 00:01:56,622
我正在用水笔解的式子

36
00:01:56,622 --> 00:01:58,245
我该如何写它呢

37
00:01:58,245 --> 00:02:00,405
所以我取对t的函数

38
00:02:00,405 --> 00:02:04,918
我有某数的sin值，其自身为关于t的函数

39
00:02:04,918 --> 00:02:07,768
所以我将在这里应用链式法则

40
00:02:07,768 --> 00:02:14,650
首先，我将求sinx的导数

41
00:02:16,508 --> 00:02:18,714
我将之写作sinx(t))

42
00:02:18,714 --> 00:02:22,365
但为了简化，我将其恢复为原式

43
00:02:22,365 --> 00:02:25,244
接着我将其与x的导数相乘

44
00:02:25,244 --> 00:02:28,766
你可以说，对于t的导数

45
00:02:28,766 --> 00:02:32,780
乘上dx/dt

46
00:02:32,780 --> 00:02:34,506
这和你之前处理链式法则相比，

47
00:02:34,506 --> 00:02:35,594
可能有些反常

48
00:02:35,594 --> 00:02:38,737
之前我们只处理与x或y相关的

49
00:02:38,737 --> 00:02:41,272
但这就是现在发生的，我将取sin外的

50
00:02:41,272 --> 00:02:43,565
对于某值的某值的导数，

51
00:02:43,565 --> 00:02:46,547
在这种情况下，即为x

52
00:02:46,547 --> 00:02:48,503
接着我将求某值的导数

53
00:02:48,503 --> 00:02:51,415
在这种情况下，即为x对于t的导数

54
00:02:51,415 --> 00:02:53,927
我们可以对第二个多项式

55
00:02:53,927 --> 00:02:56,010
用同样的方式处理

56
00:02:56,988 --> 00:03:01,216
所以我将求d/dy乘上，

57
00:03:01,216 --> 00:03:04,327
我猜你将说

58
00:03:04,327 --> 00:03:05,577
cos(y)

59
00:03:07,692 --> 00:03:09,206
接着我将之相乘：

60
00:03:09,206 --> 00:03:12,873
乘上对于t的y的导数

61
00:03:14,264 --> 00:03:17,447
接着它们整体将等于多少呢

62
00:03:17,447 --> 00:03:20,742
对于t的常数的导数

63
00:03:20,742 --> 00:03:22,162
根号2是个常数

64
00:03:22,162 --> 00:03:23,912
将不会随t的变化而变化

65
00:03:23,912 --> 00:03:27,385
所以它的导数，其变化率即为0

66
00:03:27,385 --> 00:03:29,632
好的，现在我们求到了

67
00:03:29,632 --> 00:03:31,357
所有的值

68
00:03:31,357 --> 00:03:33,681
所以第一步，sin(x)的导数为

69
00:03:33,681 --> 00:03:38,277
cos(x)乘x对于t的导数

70
00:03:38,277 --> 00:03:40,270
我将在这写下

71
00:03:40,270 --> 00:03:42,207
x对于t导数

72
00:03:42,207 --> 00:03:44,964
接着我们将有，这里是加上

73
00:03:44,964 --> 00:03:47,157
y对于t的导数

74
00:03:47,157 --> 00:03:51,010
所以加上y对于t的导数

75
00:03:51,010 --> 00:03:52,445
我在这把顺序调换了下

76
00:03:52,445 --> 00:03:54,467
所以它将提前

77
00:03:54,467 --> 00:03:58,372
现在，对于y的cos(y)的导数是多少呢？

78
00:03:58,372 --> 00:04:01,336
其值为-sin(y)

79
00:04:01,336 --> 00:04:05,265
接着，让我先把sin(y)写在这

80
00:04:05,265 --> 00:04:07,118
再加个负号

81
00:04:07,118 --> 00:04:10,118
将这（符号）擦除并换为减号

82
00:04:11,600 --> 00:04:15,100
而这依旧将等于0

83
00:04:16,062 --> 00:04:18,743
因此我们可以求出什么呢

84
00:04:18,743 --> 00:04:21,785
它告诉我们x对于t的导数的值为5

85
00:04:21,785 --> 00:04:25,398
他就在这告诉了我们

86
00:04:25,398 --> 00:04:27,481
所以这将等于5

87
00:04:29,088 --> 00:04:32,679
我们想求出对于t的y的导数的值

88
00:04:32,679 --> 00:04:36,145
他告诉了我们y的值，π/4

89
00:04:36,145 --> 00:04:40,312
这里，y是π/4，所以写下π/4

90
00:04:41,606 --> 00:04:43,791
所以我们看一下，我们得求出它

91
00:04:43,791 --> 00:04:45,725
我们仍有两个未知量

92
00:04:45,725 --> 00:04:47,449
我们不知道x的值，我们也不知道

93
00:04:47,449 --> 00:04:49,580
y对于t的导数值

94
00:04:49,580 --> 00:04:51,101
那就是我们所需求的值

95
00:04:51,101 --> 00:04:52,467
所以x的值是多少呢？

96
00:04:52,467 --> 00:04:55,420
当y等于π/4时，x的值是多少呢

97
00:04:55,420 --> 00:04:56,439
好的，为了求出它，

98
00:04:56,439 --> 00:05:00,222
我们将回到原式

99
00:05:00,222 --> 00:05:03,754
所以当y等于π/4时，你会得到

100
00:05:03,754 --> 00:05:04,847
让我把它写下

101
00:05:04,847 --> 00:05:05,680
sin(x)，加上

102
00:05:07,391 --> 00:05:08,808
cos(π/4),

103
00:05:10,716 --> 00:05:13,984
（cos(π/4)）的值为二分之根号二

104
00:05:13,984 --> 00:05:15,901
cos(π/4)

105
00:05:17,593 --> 00:05:20,936
我们可以转化为我们（熟悉的角度）或用单位圆解决

106
00:05:20,936 --> 00:05:22,558
我们正处于第一象限中，

107
00:05:22,558 --> 00:05:24,160
如果我们以角度制思考，它的值为45°

108
00:05:24,160 --> 00:05:28,067
它将等于二分之根号二

109
00:05:28,067 --> 00:05:30,579
我们可从等式两侧同时减去二分之根号二，

110
00:05:30,579 --> 00:05:32,852
我们将的值

111
00:05:32,852 --> 00:05:37,709
sin(x)的值为，好的，如果你从根号二中减去

112
00:05:37,709 --> 00:05:39,469
二分之根号二，

113
00:05:39,469 --> 00:05:40,764
你将减去它的一半

114
00:05:40,764 --> 00:05:42,223
所以你仍留有它的一半

115
00:05:42,223 --> 00:05:44,709
所以就是二分之根号二

116
00:05:44,709 --> 00:05:48,718
所以，x的值是多少呢，我将取它的sin值

117
00:05:48,718 --> 00:05:50,768
记住，这是在角度制中

118
00:05:50,768 --> 00:05:52,360
如果我们用单位圆思考，它将在第一象限。

119
00:05:52,360 --> 00:05:54,775
在这情况下x将是一个角，

120
00:05:54,775 --> 00:05:56,085
就在这

121
00:05:56,085 --> 00:05:59,376
所以那将是又一次的π/4

122
00:05:59,376 --> 00:06:03,157
所以x的值为π/4

123
00:06:03,157 --> 00:06:05,829
当y等于π/4时

124
00:06:05,829 --> 00:06:09,475
我们知道了这也是π/4

125
00:06:09,475 --> 00:06:11,437
所以让我把它重写一遍

126
00:06:11,437 --> 00:06:13,463
因为这有点乱了

127
00:06:13,463 --> 00:06:15,630
所以我们得到了五倍的

128
00:06:17,521 --> 00:06:19,354
π/4的cos值，

129
00:06:22,214 --> 00:06:23,047
减去，dy/dt，

130
00:06:24,268 --> 00:06:26,968
y对于t的导数

131
00:06:26,968 --> 00:06:28,767
我们刚求出它的值

132
00:06:28,767 --> 00:06:31,017
乘上π/4的sin值

133
00:06:32,562 --> 00:06:33,979
得0

134
00:06:35,398 --> 00:06:38,523
让我们在这里打上括号，

135
00:06:38,523 --> 00:06:40,714
以方便区分这些式子

136
00:06:40,714 --> 00:06:43,454
好的，现在我们来看

137
00:06:43,454 --> 00:06:45,108
现在只是一些简单的代数了

138
00:06:45,108 --> 00:06:46,927
cos(π/4)

139
00:06:46,927 --> 00:06:49,677
我们知道其值为二分之根号二

140
00:06:49,677 --> 00:06:53,844
sin(π/4)的值也是二分之根号二

141
00:06:54,754 --> 00:07:00,878
好现在我们看，如果同时从等式两侧除去二分之根号二会如何呢

142
00:07:00,878 --> 00:07:02,239
好的，那将告诉我们什么呢

143
00:07:02,239 --> 00:07:08,179
二分之根号二除以二分之根号二

144
00:07:08,179 --> 00:07:10,035
将等于1

145
00:07:10,035 --> 00:07:11,283
二分之根号二除以二分之根号二

146
00:07:11,283 --> 00:07:13,232
将等于1

147
00:07:13,232 --> 00:07:15,495
接着0除以二分之根号二

148
00:07:15,495 --> 00:07:17,751
也将等于0

149
00:07:17,751 --> 00:07:19,842
整个式子将简化为

150
00:07:19,842 --> 00:07:23,070
5乘1，就是5

151
00:07:23,070 --> 00:07:28,044
减去y对于t的导数，将等于0

152
00:07:29,558 --> 00:07:30,732
现在你求出了它的值

153
00:07:30,732 --> 00:07:33,648
你在等式两边同时加上y对于t的导数

154
00:07:33,648 --> 00:07:37,815
接着我们就得到y对于t的导数的值为5

155
00:07:38,684 --> 00:07:42,294
当其他条件都满足时

156
00:07:42,294 --> 00:07:44,656
即当x对于t的导数为5，

157
00:07:44,656 --> 00:07:50,166
且y的值为π/4时