And what's fallen out when we do that, because in each case, one of the first derivatives gives us the entropy.

当我们这样做时就得到了结果，因为在这些例子中，一阶导数是熵。

麻省理工公开课 - 热力学与动力学课程节选

If you knew only the third derivative of the function, you can have something quadratic in t without changing the outcome.

如果方程里有三阶导数,你就可以引入一个二次项,但是结果却不会变

耶鲁公开课 - 基础物理课程节选

So if we differentiate this object, I'm gonna find a first order condition in a second.

想要求它的导数,先让我想想一阶条件

耶鲁公开课 - 博弈论课程节选

So the slope of the guess is the first derivative.

因此斜率等于此处的一阶导数。

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So I'm hoping you guys are comfortable with the notion of taking one or two or any number of derivatives.

我希望你们,能习惯求一阶,二阶,或者任意阶导数的概念

耶鲁公开课 - 基础物理课程节选

I think the main reason is that there are no equations that involve the third derivative explicitly.

我想主要原因是因为还没有等式,明显涉及三阶导数

耶鲁公开课 - 基础物理课程节选

Normally, I will give you a function and tell you to take any number of derivatives.

通常情况下,我会给你一个函数,然后让你求任意阶的导数

耶鲁公开课 - 基础物理课程节选

The third derivative, unfortunately, was never given a name, and I don't know why.

遗憾的是三阶导数没有专门的名称,我也不知道这是为什么

耶鲁公开课 - 基础物理课程节选

Of course, you can take a function and take derivatives any number of times.

当然,你可以随意拿一个函数,对它求任意阶的导数

耶鲁公开课 - 基础物理课程节选

The first one is velocity, the second one is acceleration.

一阶导数叫速度,二阶导数叫加速度

耶鲁公开课 - 基础物理课程节选

Interestingly, only the first two derivatives have a name.

有趣的是,只有前两阶导数有名字

耶鲁公开课 - 基础物理课程节选

Once you can take one derivative, you can take any number of derivatives and the derivative of the velocity is called the acceleration, and we write it as the second derivative of position.

只要你能求一阶导数,你就能求任意阶的导数,速度的导数被称为"加速度",我们把它写成位移的二阶导数

耶鲁公开课 - 基础物理课程节选

So I differentiated this object, this is my first derivative and I set it equal to 0 Now in a second I'm going to work with that, but I want to make sure i'm going to find a maximum and not a minimum, so how do I make sure I'm finding a maximum and not a minimum?

这样我就对它求出导数了,这是一阶导数,令它等于0,一会我们就要计算了,但我先确定一下是最大值还是最小值,我怎么确定是最大值还是最小值呢

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Well, unfortunately, we know this is not the right answer, because if you take the first derivative, I get 2t.

遗憾的是,我们知道这还不是正确答案,因为如果对它求一阶导数,会得到2t

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I differentiate a second time and check the sign, so the second order condition, I differentiate this expression again with respect to q1.

我们对它进行二次求导然后看符号,这个式子的二阶导数,就是一阶导数再对q1进行求导

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Once you've got that, one derivative will give you the velocity, then in a crunch you can eliminate t and put it into this formula.

一旦你得到了这个,求一阶导数就能得到速度,然后你可以消去t,把它代入这个式子

耶鲁公开课 - 基础物理课程节选

Then, to find the meaning of b, we take one derivative of this, dx/dt, that's velocity as a function of time, and if you took the derivative of this guy, you will find as at+b. That's the velocity of the object.

接下来,为了弄清b的含义,我们取它的一阶导数,dx/dt,得到速度作为时间的函数,如果你对它求导的话,你会得到at+b,这就是物体的速度

耶鲁公开课 - 基础物理课程节选