So, all I want to do now is look at the derivatives of the free energies with respect to temperature and volume and pressure.

我现在所要做的一切就是,考察自由能对,温度，体积和压强的偏导数。

麻省理工公开课 - 热力学与动力学课程节选

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

That is, it's easy to write down straight away that dG with respect to temperature at constant pressure S is minus S.

这就是说，可以很简单的写出dG在,恒定压强下对温度的偏导数,是负。

麻省理工公开课 - 热力学与动力学课程节选

And, coincidently, what is the partial of energy with respect to distance?

巧合的是，能量对距离的导数是什么？

麻省理工公开课 - 固态化学导论课程节选

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

耶鲁公开课 - 博弈论课程节选

Let's sneak in one more derivative here, which is to take the derivative of the derivative.

我们再求一次导数,也就是对导数求导

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

You can actually take derivatives of a vector with time.

你可以直接求位矢对时间的导数

耶鲁公开课 - 基础物理课程节选

And then we can take the derivative with respect to temperature, it's just R over molar volume minus b.

这样我们求,压强对温度的偏导数，结果等于R除以摩尔体积V杠减去b的差。

麻省理工公开课 - 热力学与动力学课程节选

So, using those, now, what happens if we take the second derivative of A, the mixed derivative, partial with respect to T and the partial with respect to V.

如果我取A的二阶导数,混合导数，对T偏微分,再对V偏微分。

麻省理工公开课 - 热力学与动力学课程节选

Just take the derivatives and that's the velocity vector.

直接计算对时间的导数就是速度矢量

耶鲁公开课 - 基础物理课程节选

And the mathematics of that equation involved a double derivative in time of x 0 plus some constant times x equals zero with some constraints on it.

那个数学方程式,包括了x对时间的二阶导数,加上常数乘以x等于,还有一些限制条件。

麻省理工公开课 - 固态化学导论课程节选

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

耶鲁公开课 - 博弈论课程节选

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

耶鲁公开课 - 基础物理课程节选

With respect to n, the number of moles.

对n的偏导数，这里的n是摩尔数。

麻省理工公开课 - 热力学与动力学课程节选

You can take a derivative of the derivative and you can get the acceleration vector, will be d^2r over dt^2, and you can also write it as dv over dt.

你可以对导数再求一次导,你就可以得到加速度矢量,也就是 d^2 r / dt^2,你也可以写成 dv / dt

耶鲁公开课 - 基础物理课程节选

G We can take the derivative of G with respect to how much material there is.

我们可以取,对物质总量的偏导数。

麻省理工公开课 - 热力学与动力学课程节选

So d/dT of dA/dV, just like this.

即对dA/dV求对温度T的偏导数。

麻省理工公开课 - 热力学与动力学课程节选

And the point is that the second derivative of A, with respect to V and T in this order is the same as the second derivative of a with respect to T and V in this order.

问题的关键在于A的二阶导数，对V和T以这样一个顺序求导,和对T和V以这样一个顺序,求导是一样的。

麻省理工公开课 - 热力学与动力学课程节选

What I did was I differentiated this fairly simple function with respect to q1 and since I want to find a maximum, what I'm going to do is I'm going to set this thing equal to 0.

只不过是对q1求导了,既然我们要求出最大值,只需要令导数等于0就可以了

耶鲁公开课 - 博弈论课程节选