Is ∂x/∂f = 1/(∂f/∂x) a Valid Equation in Implicit Differentiation?

[aaaa202]

I have some complicated function f of the variables x,y:

f(x,y)

Now I can't really invert this expression for f for x and y, but I want the derivative of x and y wrt f. How can I do that? Am I allowed to say:

∂x/∂f = 1/(∂f/∂x)

I have seen physicists "cheat" by using this relation, though I am not sure that it is always true. In general when can I do the above?

 

[BvU]

I take it f is a number, not a vector ? That is difficult to invert (making two variables out of one function value?), so it will also be difficult to get those derivatives!

When I don't understand things, I try an example. Here I try $f = x^2+y^2$ and the best I can do is write
$$df = \left(\partial f\over \partial x \right)_y dx + \left(\partial f\over \partial y \right)_x dy$$ Physicists 'cheat' as much as they can get away with -- just like everybody else. There steno jargon does get misquoted frequently, though. Can you give an example ? Maybe we can work out what they meant, but wrote down in a sloppy shortcut sort of way...

 

[Chestermiller]

I have some complicated function f of the variables x,y:

f(x,y)

Now I can't really invert this expression for f for x and y, but I want the derivative of x and y wrt f. How can I do that? Am I allowed to say:

∂x/∂f = 1/(∂f/∂x)

I have seen physicists "cheat" by using this relation, though I am not sure that it is always true. In general when can I do the above?

Always. The partials in the equation imply that y is being held constant. In that case, f is just a function of the single variable x, and x is just a function of the single variable f.

Chet