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.
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.
它是收入Pq相对于数量的导数。
It is the derivative of revenue pq with respect to quantity.
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.
但都是使用导数的逼近公式。
But, it's the usual approximation formula using the derivative.
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.
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.
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.
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.
那么dx和dy前面的系数就是偏导数。
我们首先要做的事是,求偏导数，fx是多少？
What we would start doing immediately is taking the partial derivatives. What is f sub x?
首先考虑第一个分量，求出它关于x的偏导数。
Ry-Qz Well, so we have to take the first component, Ry minus Qz, and take its partial with respect to x.
这一因素被称为是“二阶导数”，或者是变化率的变化率。
This factor has been dubbed "the second derivative", or the rate of change of the rate of change.
未知量是一个函数，这个方程，将把函数的偏导数联系起来。
So the unknown is a function, and the equation will relate the partial derivatives of that function to each other.
基本上，答案就是，我们不可以用二次导数来验证这个情况。
Basically, the answer for us is that we don't have a second derivative test in this situation.
扭矩除以转动惯量,就会引起角加速度，也就是角速度的导数。
Torque divided by moment of inertia is what will cause the angular acceleration, namely the derivative of angular velocity.
这个类似于，力除以质量等于加速度，就是速度对t的导数。
That is the analog of force divided by mass equals acceleration, which is d over dt of velocity.
这样x为自变量，z为因变量,这条曲线的斜率就是关于x的导数。
And when I change x, z changes, and the slope of that is going to be the derivative with respect to x.
这是一个临界点，考察它的二阶导数,会发现这确实是一个极大值点。
That is a critical point. And we used the second derivative to see that this critical point is a local maximum.
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.
人与数据库交互：他们引导数据流的走向；他们解释并处理数据请求。
People interact with databases: they direct data flows; they interpret and service data requests.
和以前一样，它来源于这样一个事实，混合二阶导数，求导次序改变，结果不变。
And, as before, this comes from the fact that the mixed second derivatives are the same, no matter in which order you take them.
我确信你们仍然记得一些简单的规则，并且能够迅速地告诉我x2的导数是2x。
I'm sure you remember some of the simpler rules for differentiation and could quickly tell me that the derivative of x2 is 2x.
有必要对加速度给出一个严格的定义，加速度,作为一个矢量,是速度的一阶导数。
So, the definition will take effect. The acceleration is as a vector, and the acceleration vector is just the derivative of a velocity vector.
他们说，二阶导数正在趋正，意思是：尽管经济仍在下滑，但是下滑的速度减缓了。
The second derivative, they say, is turning positive. That means that although the economy is spiralling down, it is doing so more slowly.
如果取的向量场是处处恒定的，所有点都是平移关系，所以没有散度，因为导数为零。
If you take a vector field that is a constant vector field where everything just translates then there is no divergence involved because the derivatives will be zero.
微积分基本定理，不是曲线积分的，告诉我们，如果对函数的导数积分，就会得回原函数。
So, the fundamental theorem of calculus, not for line integrals, tells you if you integrate a derivative, then you get back the function.
微积分基本定理，不是曲线积分的，告诉我们，如果对函数的导数积分，就会得回原函数。
So, the fundamental theorem of calculus, not for line integrals, tells you if you integrate a derivative, then you get back the function.
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