# Ambiguity with partial derivative notation

Suppose I have some function f that depends on three variables, namely x, y, and t; i.e.,
$$f=f(x,y,t).$$
Now suppose that y depends on x, i.e., $y=y(x)$. Taking this into account, we see that f is really just a function of two independent variables, x and t. So my question is this: if I write down the partial derivative of f with respect to x, i.e.,
$$\frac{\partial f}{\partial x},$$
it seems to me that there is ambiguity in whether y is to be held constant or allowed to vary. Perhaps to make it clear we ought to write
$$\frac{\partial f(x,y,t)}{\partial x}, \quad \frac{\partial f(x,y(x),t)}{\partial x};$$
the first one means that y is held constant in the differentiation, while the second one means that y should be treated as dependent on x.
It would obviously be wrong to use total derivative notation with straight d's for the second case since there are still two independent variables, x and t.

Is the notation I have used the correct way of distinguishing the two cases, or is there some other way?

I haven't been able to find any info on this question on the internet so thought i might be worth asking :).

Last edited:

tiny-tim
Homework Helper
hi perishingtardi! welcome to pf! if there was no t, ie f(x,y),

then the first would be written ∂f/∂x, and the second df/dx​

however, the t stops us doing that,

so i think we have to define a new g(x,t) = f(x,y(x),t),

and then the second is ∂g/∂x thanks tiny-tim, i didnt think of that trick although it seems quite obvious now! :)

I think have seen the notation $(\frac{ ∂f }{ ∂x })_{y}$, especially in thermodynamics equations where a lot of variables are interdependent. Here the variables in subscript are held constant.