Derivatives of a Function: General Case

[Physics_wiz]

Say I have a function $$g = g(x,y)$$. Now, I have another function defined as:
$$f(x,y) = \partial / \partial x [g(x,y)]$$. Is the following true:

$$f(a,b) = (\partial / \partial x [g(x,y)])_{x = a, y = b} = \partial / \partial a [g(a,b)]$$

a, b, x, y are all variables. Is this true in general for any function of any number of variables?

 

[dhris]

It looks like you just substituted a for x and b for y. So I guess the answer is yes, you can do that :).

 

[neutrino]

$$f(a,b) = (\partial / \partial x [g(x,y)])_{x = a, y = b}$$

To me that looks like the partial derivative of g with respect to x, at the point (a,b). It's better without that middle part, if you just want to give the variables a different set of 'labels'.

 

[Physics_wiz]

To me that looks like the partial derivative of g with respect to x, at the point (a,b). It's better without that middle part, if you just want to give the variables a different set of 'labels'.

Yes, you're right in your interpretation. However, the equality between the middle part and the last part is the most crucial step for my purposes, so I can't take it out.

 

[HallsofIvy]

The only difference between the "middle" and "last" parts of you statement are that in the middle part, you differentiate using the "symbols" x and y, then replace them by a and b, while, in the last part, you replace x and y by a and b, then differentiate. Yes, those are equal.