What's the difference between $\frac{\partial f}{\partial x}(x,y)$ and $\frac{\partial f (x,y)}{\partial x}$?

Question

Is there a difference in the meaning of $\frac{\partial f}{\partial x}(x,y)$ and$\frac{\partial f (x,y)}{\partial x}$ or is it just a different notation for exactly the same thing?

Answer 1

$$\frac{\partial f (x,y)}{\partial x}$$

$$\frac{\partial f}{\partial x}(x,y)$$

They both mean the same.

The notation for a partial derivative depends on what we want to emphasize. For example

$\frac{\partial f}{\partial x}(x_0,y_0)$ Partial derivative of f with respect to x at $(x_0, y_0)$ Convenient for stressing the point.

$\frac{\partial z}{\partial x}|_{(x_0,y_0)}$ common in science and engineering when you are dealing either variables and do not mention the function explicitly.

Answer 2

It's exactly the same thing. I believe the former emphasizes the fact that the derivative of $f$ is a function of $x$ and $y$ which is also implicit in the latter symbol because $f$ is written as a function of $x$ and $y$.