我们再求一次导数，也就是对导数求导。
Let's sneak in one more derivative here, which is to take the derivative of the derivative.
这些需求导致了高等教育体系的迅速扩张，它始于20世纪60年代，迅速发展于70年代和80年代（尽管并不均衡）。
These demands resulted in a very rapid expansion of the systems of higher education, beginning in the 1960s and developing very rapidly (though unevenly) during the 1970s and 1980s.
这里就需要知道如何对倒数求导。
事实上，可以把求导放在积分里面。
现在我们要对这个量的平方求导。
And now we have to take the derivative of this quantity squared.
那么乘积法则对点乘求导没问题。
So, product rule is OK for taking the derivative of a dot product.
扩大的需求导致教学资源紧张。
最开始我讲过了，如何对倒数函数求导。
What I showed you at the beginning, and then erased, basically justifies how to take the derivative of a reciprocal function.
通常，最糟的技术问题由非功能需求导致。
Commonly, the toughest technical problems are driven by non-functional requirements.
以通常的方式求导。
但是，当我不想替换的时候，我可以先求导数。
And when I don't really want to actually substitute arctangents everywhere maybe I would rather deal with the derivatives.
这和一般的除法求导法则，仅在写法上有点不同。
And that is actually the usual quotient rule just written in a slightly different way.
对燃料的需求导致艾伯丁裂谷地区的森林日渐稀薄。
Demand for fuel is thinning forests in the Albertine Rift region.
对接受BEC培训的高需求导致了网络的迅速扩大。
High demand has resulted in the rapid enlargement of the network.
竟然可以存在一个直接的等式，没有任何求导的步骤。
And you can get a direct equality without any derivatives or anything.
所以得到温度微小变化的一种方法，就是对温度求导。
So one way to get a small change in temperature is to take a derivative with respect to temperature.
如果说你要知道，关于一元微积分中所有的求导法则。
Let's say that you want to understand actually all these rules about taking derivatives in single variable calculus.
日益增长的对廉价食物的需求导致了另一种形式的污染。
The need to produce ever-increasing quantities of cheap food leads to a different kind of pollution.
对这个方程两边对总的压强求导，好，这里不是总压强。
So let's take the derivative of this with respect to total pressure. Well, this isn't total pressure.
然后求导。
接下来我想知道，对于多个互有联系的变量，该如何求导。
Then I would want to know maybe how the derivatives, with respect to the various sets of variables, related to each other.
对压强求导，你知道吉布斯自由能，的变量是压强和温度？
With respect to pressure, when we know that the variables that we use for Gibbs free energy are pressure and temperature. Right?
我的意思是加速度只沿这一个方向，它只要对速率求导即可。
I mean, acceleration is also going to be along the line, and it's going to has to do with the derivative of speed.
另一方面，可以从对y积分开始,然后关于x求导，进行匹配。
I mean, on the other hand, you could start with integrating with respect to y and then differentiate and match with respect to x.
这种需求导致了敏捷联盟的成立，以及第一次敏捷大会的组织工作。
This need led to the establishment of the agile Alliance and then to the organization of the first agile conference.
对复杂按揭债券的需求导致了信贷标准的放松，进而把房价抬得更高。
Demand for complex mortgage securities led to a loosening of lending standards, which in turn drove house prices higher.
如果只有一个变量，要求关于这个变量的偏导，这和通常求导是一样的。
If you have a single variable then the partial, with respect to that variable, is the same thing as the usual derivative.
过去几年，由于中国的强劲需求导致市场供应吃紧，铁矿石价格大幅上涨。
Iron ore prices have risen substantially over the last few years amid tight supplies because of strong demand from China.
还记得吧，对一个量的平方求导，就得到这个量的2倍，乘以这个量的导数。
Remember, we take the derivative of the square. We take twice this quantity times the derivative of what we are squaring.
如果你对它关于x求导,就会得到2xy，而这项就没有了,因为它里面没有。
See, if you took the derivative of this with respect to x you will get 2xy and this guy will go away x because there is no x in it.
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