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.
那么，那个方向的导数是什么呢？
它只有关于每个变量的偏导数。
它其实是那个方向的方向导数。
Well, it's actually the directional derivative in that direction.
它就是位置，对时间的一阶导数。
角加速度等于，角速度的导数。
And angular acceleration is the derivative of angular velocity.
基本上是的，那就是方向导数。
And that's basically, yes, that's the directional derivative.
那么二阶导数判定是怎样的呢？
我是说现在已经有两个一阶偏导数。
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.
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.
它是收入Pq相对于数量的导数。
It is the derivative of revenue pq with respect to quantity.
但都是使用导数的逼近公式。
But, it's the usual approximation formula using the derivative.
好的，那就是偏导数的定义。
垂直于梯度的方向上,方向导数为零。
The directional derivative in a direction that's perpendicular to the gradient is basically zero.
他们说二阶导数转为正值了。
总而言之，我们所做的就是求二阶导数。
我们还要试图理解偏导数。
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.
速度向量，是位置向量关于时间的导数。
So, the velocity vector is the derivative of a position vector with respect to time.
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.
不是每个函数都有导数。
因此，一个多变量的函数没有通常的导数。
So, a function of several variables doesn't have the usual derivative.
那就是，每个小盒子里烟雾总量的导数和。
Well, that will be the sum of the derivatives of the amounts of smoke inside each little box.
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.
我们还学过加速度，也就是速度向量的导数。
And, we've also learned about acceleration, which is the derivative of velocity.
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.
对于任意两分量，混合偏导数相等。
For every pair of components the mixed partials must be the same.
我会计算时间导数。
临界点是，偏导数都为零的点。
So, critical points, remember, are the points where all the partial derivatives are zero.
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