Coursera | Andrew Ng (01-week-2-2.5)

该系列仅在原课程基础上部分知识点添加个人学习笔记，或相关推导补充等。如有错误，还请批评指教。在学习了 Andrew Ng 课程的基础上，为了更方便的查阅复习，将其整理成文字。因本人一直在学习英语，所以该系列以英文为主，同时也建议读者以英文为主，中文辅助，以便后期进阶时，为学习相关领域的学术论文做铺垫。- ZJ

Coursera 课程 |deeplearning.ai |网易云课堂

转载请注明作者和出处：ZJ 微信公众号-「SelfImprovementLab」

知乎：https://zhuanlan.zhihu.com/c_147249273

CSDN：http://blog.csdn.net/JUNJUN_ZHAO/article/details/78891826

Derivatives(导数)
(字幕来源：网易云课堂)

In this video I want to help you gain an intuitive understanding of calculus and the derivatives.now maybe you’re thinking that you haven’t seen calculus since your college days,and depending on when you graduate,maybe that was quite some time back.now if that’s what you’re thinking don’t worry, you don’t need a deep understanding of calculus,in order to apply neural networks and deep learning very effectively.

在这个视频中，我想让你对微积分和导数有直观的理解。或许你认为自从大学后，你再也没有接触微积分，这取决于你什么时候毕业,也许有一段时间了，如果你顾虑这点，请不要担心，你并不需要非常深入理解微积分就能高效应用神经网络和深度学习。

so if you’re watching this video or some of the later videos be wondering,wow this stuff really for me,this calculus looks really complicated. My advice to you is,the following which is that watch the videos,and then if you could do the homework and complete the programming homework successfully,then you can apply deep learning. in fact what you see later is that in week 4 will define a couple of types of functions，that will enable you to encapsulate everything,that needs to be done with respect to calculus that these functions called forward functions and backward functions that you learn about that let you put everything you need to know about calculus into these functions,so that you don’t need to worry about them anymore beyond that.

因此，如果你观看这个视频或者之后视频时想到，哇喔，这些知识，这些运算对我来说真的很复杂。我给你的建议是，坚持学习视频，最好做好课后作业，成功完成编程作业，就能运用深度学习了。事实上，之后你将看到，在第 4 周将会定义，很多种类的函数类型，它们能够帮助你。通过微积分把所有需要的知识结合起来，其中有叫做“前向函数”，和“反向函数”的函数，你不需要了解很多微积分去使用这些函数，因此你无需担心它们超出理解范围。

but I thought that in this foray into deep learning,that this week we should open up the box and peer a little bit further into the details of calculus,but really,all you need is an intuitive understanding of this,in order to build and successfully apply these algorithms.oh and finally if you are among that maybe smaller group of people that are expert in calculus,if you’re very familiar with calculus observe this,it’s probably okay for you to skip this video.

但在深度学习中，这周我们要一探究竟，进一步深入了解微积分的细节。但其实你只需要对这些微积分有直观的认识，就可以构建和成功应用这些算法。最后，如果你是为数不多的、精通微积分的同学，如果你对微积分非常熟悉，你可以跳过这个视频。

but for everyone else let’s dive in and try to get an intuitive understanding of derivatives.I’ve plotted here the function f of a equals 3a.so it’s just a straight line to gain intuition about derivatives.let’s look at a few points on this function.let’s say that a is equal to 2,in that case f of a which is equal to 3 times a is equal to 6.so if a is equal to 2,then you know F of a will be equal to 6,let’s say we give the value of a,you know just a little bit of a nudge /nʌdʒ/ 推动 .I’m going to just bump up me a little bit,so there is now 2.00 1 right,so I’m going to get a like a tiny little nudge to the right,so now is let’s say 2.001,this plug this is to scale 2.001,the 0.001 difference is too small to show on this plot,this give them a little nudge to the right,now f of a is equal to three times a,so six point zero zero three.

其他同学我们开始吧，一起深入学习导数，我在这里画了一个函数 f(a)=3af(a)=3a 它是一条直线，直观的理解导数，让我们看看函数中几个点，假定 a=2a=2，那么f(a)f(a) 等于 aa 的 3 倍等于 6，也就是说如果a=2a=2，那么函数f(a)=6f(a)=6，我们稍微改变aa的值，只变一点点移动，只增加一点点，就到2.001，往右小小地移动一下，这时是 2.001，这个坐标值是 2.001，0.001的差别实在太小了，不能在图中显示我们把它右移一点，现在f(a)f(a) 等于aa的3倍，是6.003.

Simplot this over here,this is not to the scale this is six point,zero zero three so if you look at this low triangle here,some highlighting in green what we see is that,if I match a 0.001 to the right,then F of a goes up by 0.003 the amount that F of a went up,is three times as big as the amount that I nudged a to the right,so we’re going to say that the slope or the derivative,of the function f of a at a equals two,or when a is equal to 2 the slope is three,and you know the term derivative,basically means slope is just that,derivative sound like a scary a more intimidating word,whereas slope is a friendlier way to describe the concept of derivative.

画在这里，比例不太符合，这是，6.003 你看这个小三角形，绿色高亮部分，如果我向右移动 0.001，那么f(a)f(a)增加 0.003，ff的值增加的值，3 倍于aa往右移的量，因此我们说斜率，即导数，函数f(a)f(a) 在a=2a=2处，或者说当a=2a=2时斜率是 3，导数这个概念，基本意味着斜率，导数听起来是个很吓人的词，但是斜率以一种很友好的方式，来描述导数这个概念。

so one of these year derivative just think slope of the function,and more formally the slope is defined as the height divided by the width of this little triangle,that we have in green so this is you know 0.003,over 0.001 and the fact that the slope is equal to 3,or the derivative is equal 3 just represents the fact that,when you nudge a to the right by 0.001 by tiny amount,the amount that F of aa goes up,is three times as big as the amount that,united the united a in the horizontal direction.so that’s all that the slope of a line,is now let’s look at this function at a different point,let’s say that a is now equal to five,in that case f of a three times a is equal to 15.

所以有时提到导数把它当做函数的斜率就好了，更正式的斜率定义，为 高除以宽，在这个绿色小三角形，即 0.003，除以 0.001 斜率等于 3，或者说导数等于 3 这表示，当你将 aa 右移 0.001 移动一点点，f(a)f(a) 的值将增加，3倍的量，水平方向aa的总量，这就是这条线的斜率，现在让我们在不同点看看这个函数，假设a=5a=5，此时f(a)f(a) 3乘以aa，等于15 。

so let’s say I again give a and notch to the right,a tiny level less is now bumped up to five point oo one,F of a is three times that,so f of a is equal to fifteen point zero three,and so once again when I bump up a to the right nudge a,to the right by 0.001.F of a goes up three times as much,so the slope again at a equals five is also three,so the way we write is that,the slope of the function f is equal to three,we say df of a da,and this just means the slope of the function f of a,when you nudge the variable a a tiny little amount um,this is equal to three

我再一次右移aa，很小幅度增加到 5.001，f(a)f(a)等于 3 倍的……，f(a)f(a)等于 15.003，再一次当我增加aa，将aa右移0.001，f(a)f(a)增加 3 倍，所以在a=5a=5时斜率是 3，我们这样来写，把函数f=3f=3的斜率，写为df(a)dad \dfrac{f(a)}{da}，这表示f(a)f(a)的斜率，当你微小改变变量aa的值，它等于3。

and an alternative way to write this derivative formula is,as follows you can also write this as d da of f of a,so whether you put the f of a on top of,whether you write it you know down here,it doesn’t matter,but all those equation means is that,if I nudge a to the right a little bit.I expect F of a to go up by three times,as much as I nudge just the value of little a,now for this video I explained derivatives,talking about what happens,we nudge the variable a by 0.001 um,

导数公式的另一个的表达方式，你可以这样写 ddaf(a)\dfrac{d}{da}f(a) ，不管你是否将f(a)f(a)放在上面，或是放在这里，都没有关系，这些等式意味着，将aa右移一点，f(a)f(a)会增加 3倍于，我移动aa的值，在这个视频中我解释了导数，讨论的情况是，我们将变量aa偏移 0.001，

if you want the formal mathematical definition of the derivatives,derivatives are defined with an even smaller value of how much you nudge a to the right,so it’s not often 00 1 is not 0.00000001,is not 0.000000 and so on 1 is sort of,even smaller than that,and the formal definition of derivative,says what have you nudge a to the right by an info testable amount ,basically an infinite infinitely tiny tiny amount,if you do that just f of a,go up three times as much as whatever,was a tiny tiny tiny amount that you now stay to the right.

如果你想知道正式的导数数学定义，导数是,你右移aa，非常小的值，不是 0.001，不是 0.00000001 不是 0.00000..1，非常非常小的值，导数的定义是，你右移aa 一个不可度量的值，一个无限小的值，f(a)f(a)会增加，增加了一个非常非常小的值的 3 倍，也就是在这个三角形右边的变化值。

【wiki | 导数】:https://zh.wikipedia.org/wiki/%E5%AF%BC%E6%95%B0

so that’s actually the formal definition of a derivative,but for the purposes of our intuitive understanding,we’re going to talk about nudging a to the right by this small amount 0.001,even if it’s 0.001,isn’t exactly you know tiny tiny insa testable,now one property of the derivative,is that no matter where you take the slope of this function,it is equal to 3 whether a is equal to 2 or a,is equal to 5 the slope of this function,is equal to 3 meaning that,whatever is the value of a if you increase it by 0.001,that value of f of a goes up by three times as much,so this function has the same slope everywhere,and one way to see that is that wherever you draw this your,little triangle right the height divided,by the width always has a ratio of three to one.

那就是导数的正式定义，但是为了直观地认识，我们将探讨，右移aa以0.001 这个值，尽管 0.001，并不是无穷小的可测数据，导数的一个特性是，这个函数任何地方的斜率，总是等于 3 不管 a=2a=2或a=5a=5，这个函数的斜率，总等于 3 就是说，不管aa的值如何如果你增加 0.001，f(a)f(a)的值就增加 3 倍，这个函数在所有地方的斜率都相等，一种证明的方式是无论你将小三角形，画在哪里它的高除以宽的比值，总是 3 比 1 。

so I hope this gives you a sense,of what the slope what the derivative of the function,means for a straight line,where in this example the slope of the function was three everywhere in the next video.let’s take a look at a slightly more complex example,where the slopes of the function can be different at different points on the function.

我希望给你一种感觉，什么是斜率，什么是导数，对于一条直线而言，在例子中,函数的斜率，在任何地方都是 3，在下一个视频中，我们会看一个更复杂的例子，那个函数的斜率在不同点处，都是不一样的。

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