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

What's a function, what's a derivative, what's a second derivative, how to take derivatives of elementary functions, how to do elementary integrals.

什么是函数,什么是导数,什么是二阶导数,如何对初等函数求导,如何进行初等积分

耶鲁公开课 - 基础物理课程节选

One way to think about this intuitively if the derivative is very large the function is changing quickly, and therefore we want to take small steps.

关于这个方法很直观的一点想法是,如果导数非常大,函数也就变化的非常快，因此我们想一小步一小步的来。

麻省理工公开课 - 计算机科学及编程导论课程节选

I tell you something about the second derivative of a function and ask you what is the function.

我告诉你一个函数的二阶导数,然后问你这个函数是什么

耶鲁公开课 - 基础物理课程节选

Change the independent variable, find the change in the function, take the ratio and that's the derivative.

改变自变量,算出函数的变量,计算比值,这就是求导

耶鲁公开课 - 基础物理课程节选

If I give you a function, you know how to take the derivative.

如果我给你一个函数,你就知道如何求导

耶鲁公开课 - 基础物理课程节选

This function will have the right second derivative.

这个函数会得出正确的二阶导数

耶鲁公开课 - 基础物理课程节选

Everyone knows from calculus that if you're trying to find a function about which you know only the derivative, you can always add a constant to one person's answer without changing anything.

学过微积分的人都知道,如果你想根据已知的导数,求出其原函数,你总是可以给某人的答案,随便加一个常数,且不影响结果

耶鲁公开课 - 基础物理课程节选

In other words, your job is to guess a function whose second derivative is a, and this is called integration, which is the opposite of differentiation, and integration is just guessing.

换言之,你的任务是要猜出一个二阶导数为a的函数,这就是积分,和微分恰恰相反,积分就是猜

耶鲁公开课 - 基础物理课程节选

Mathematically complete problem is that you can find the function x of t by saying that the second derivative of the function is equal to -k over m times the function.

这个数学问题就是,你可以求出函数 x,只要令函数的二阶导数,等于 -k 除以 m 再乘以函数

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

Now, I hope you guys know that much calculus, that when you take a derivative of a function of a function, namely v square over 2 is a function of v, and v itself is a function of t, then the rule for taking the derivative is first take the v derivative of this object, then take the d by dt of t, which is this one.

我希望你们了解更多的微积分知识,当你对复合函数求导时,也就是说v^/2是关于v的函数,而v本身是关于t的函数,求导的法则应该是,第一步是这一部分对v求导,然后v再对t求导,得到这一部分

耶鲁公开课 - 基础物理课程节选