那么，那个方向的导数是什么呢？

And then, what's the derivative in that direction?

youdao

它只有关于每个变量的偏导数。

It has only partial derivatives for each variable.

youdao

它其实是那个方向的方向导数。

Well, it's actually the directional derivative in that direction.

youdao

它就是位置，对时间的一阶导数。

That's the first derivative of the position versus time.

youdao

那么二阶导数判定是怎样的呢？

So, what does the second derivative test say?

youdao

我是说现在已经有两个一阶偏导数。

I mean we already have two first derivatives.

youdao

有些不是规则的函数，却有导数。

You have functions that are not regular enough to actually have a derivative.

youdao

我们要求这个，方程的时间导数。

And so we're going to take the derivative versus time of this equation.

youdao

比较这两个，我们会得到h的导数。

Now, if we compare these two, we will get the derivative of h.

youdao

好的，那就是偏导数的定义。

OK, so that's the definition of a partial derivative.

youdao

他们说二阶导数转为正值了。

The second derivative, they say, is turning positive.

youdao

你们可能认为，加速度只是速度的导数。

So, you might think acceleration is just the derivative of speed.

youdao

总而言之，我们所做的就是求二阶导数。

And all we did, further, is take that second derivative.

youdao

我们还要试图理解偏导数。

We are trying to understand partial derivatives.

youdao

因此，我们应该重新理解偏导数的含义。

So, we have to figure out what we mean by partial derivatives again.

youdao

当遇到偏导数时，一定记住，不能约分。

Somehow, when you have a partial derivative, you must resist the urge of simplifying things.

youdao

不是每个函数都有导数。

And, not every function has a derivative.

youdao

把对压强的导数拿出来，看看有什么发生。

Let's take the derivative outside with pressure and see what happens.

youdao

我们已经发现,它由f对x的偏导数给出。

Well, we have seen that it is given by the partial derivative f sub x.

youdao

我们来用方向导数，来描述一下相同的东西。

Let's say the same thing in terms of directional derivatives.

youdao

我会计算时间导数。

I take the time derivative.

youdao

我只包括在未来一段导数的定义。

I will just include a definition of the derivative in the next section.

youdao

对n的偏导数，这里的n是摩尔数。

With respect to n, the number of moles.

youdao

它的二阶导数就是这个恒加速度。

The second time derivative is the constant acceleration.

youdao

与磁场的耦合是通过长导数得到的。

The coupling to magnetic field is via the long derivative.

youdao

因此，在教学中，要突出导数的应用。

Therefore, in the teaching, must highlight the derivative the application.

youdao

同样地，梯度也是把偏导数糅合到一起。

Well, the gradient is also a way to package partials together.

youdao

但我们仍想问，是否在每个方向上都有一个导数呢？

But, we can still ask, is there a derivative in every direction?

youdao

我们不可以用2阶导数来观察或者别的什么方法。

We cannot use second derivative tests or anything like that.

youdao

想象一下，一个导数作为车上的车速计。

Imagine a derivative as the speedometer on a car.

youdao