Derivatives with quantities held constant

Question

I am running into the following notation:

$$\left(\frac{\partial x}{\partial y}\right)_w$$

which is stated to be "the partial derivative of $x$ with respect to $y$ at constant values of $w$" and where

$$x=x(y,w).$$

(I see this in physics literature, for example when dealing with entropy and adiabaticity.)

I am just wondering how I can calculate this kind of quantity. I have quantities $x,y,w$ at times $t-1, t, t+1$ etc and want to calculate that derivative at time $t$. How can I do it?

Answer 1

You seem to have a function $x = x(y, \ldots)$ so your expression requires you to compute $$ f(y, \ldots) = \frac{\partial x}{\partial y} $$ and then evaluate $f(w, \ldots)$. I think the corresponding notation in mathematics books would be $$ \left. \frac{\partial x}{\partial y} \right|_{y=w} $$