Why use der-f instead of df in partial derivatives?

[marschmellow]

I understand the difference between a differential and a partial differential--at least I think I do. A partial differential represents a tiny change in a variable when all other variables are held constant, a differential represents a tiny change in a variable when all other variables may or may not be constant. For this reason it makes a lot of sense to use der-x or der-y when taking partial derivatives, because you don't want possible changes in other variables to screw with how one variable changes the value of the function.

But why do we use a partial differential on the function itself? It seems like you could just write df/der-x and you'd be fine. Since x is the only thing changing already, the function can only change in one way.

Thanks

 

[hotvette]

You can thank Gottfried Leibniz (1646-1716) for the notation we use:

'Leibniz occupies a prominent place in the history of mathematics and the history of philosophy. Leibniz developed the infinitesimal calculus independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published'

 

[Office_Shredder]

Is your question why do we write $$\frac{\partial f(x,y)}{\partial x}$$ instead of $$\frac{df(x,y)}{dx}$$?

 

[trambolin]

No, the question is why not writing $$\frac{df(x,y)}{\partial x}$$ instead of having both terms with $$\partial$$. Since the change in the function is independent of what is being changed. I think that's a good question and I don't know why.

 

[HallsofIvy]

It's really just a matter of convention. You could replace every $\partial f/\partial x$ in a textbook with $df/dx$ (or $\partial f/dx$ although I can think of no good reason for that mixed notation) and, as long as you remember that f is a function of more than one variable, there would be no difference.

 

[lugita15]

As it stands, $$\partial f$$ has no meaning by itself, unlike in the single variable case where $$df = f(x+dx)-f(x)$$.
I think a more logical notation would be $$\partial_{x} f = f(x+dx,y)-f(x,y)$$ and $$\partial_{y} f = f(x,y+dy)-f(x,y)$$. Then the partial derivatives of f would be $$\frac{\partial_{x} f}{dx}$$ and $$\frac{\partial_{y} f}{dy}$$. Also, the multivariable version of the chain rule would become obvious, just like the single variable version is. You just use the definition $$df = f(x+dx,y+dy)-f(x,y)$$, and you'd get $$df = \partial_{x} f + \partial_{y} f$$. So the infinitesimal change in f is the change if you hold x constant plus the change if you hold y constant.

But unfortunately, the existing notation is too well-established, and will likely never change.