Formalizing differation of a function $f$ by function $g$ Sometimes (usually in physics) i encounter expressions such as $\dfrac{d}{dg(x)} f(x)$ 
This expression is quite intuitive: what is the change in $f(x)$ with respect to change in $g(x)$?
How can i formalize this in terms of limits? Is it the same as:
$$\dfrac{d}{dg(x)} f(x)\big|_{x=x_0} = \lim_{g(x)\rightarrow g(x_0)}\dfrac{f(g(x))-f(g(x_0))}{g(x)-g(x_0)} $$
If so, this looks like the "usual" differation of composite function by $x$, but i cant really figure out the expression $g(x)\rightarrow g(x_0)$. 
-How does $g(x)$ tend to $g(x_0)$?
-Does $g(x)$ need to be 1-1 map?
For example, if we look at $r:\mathbb R\rightarrow \mathbb R $ defined by: $r(t) = \sqrt{x^2(t)+y^2(t)+z^2(t)}$

What is $\dfrac{dr(t)}{d\dot r(t)}$?
 A: Whenever such an expression occurs, $f(x)$ can in fact be expressed as the composition of a function $h$ and the function $g$ :
$$f(x)=h(g(x))$$
In that case, what you write $\dfrac{d}{dg(x)}f(x)$ could more properly be written $\dfrac{d}{dy}h(y)\vert_{y=g(x)}$. 
The variable appearing in the $\dfrac{d}{dy}$ must be an independent variable, in order to ensure the definition in terms of limits : here you have $\lim\limits_{y\rightarrow g(x_0)}$.
For this reason, I don't think one could give a reasonable meaning to your example with $r(t)$, since $r(t)$ is not a function of $r'(t)$. In fact, if $r(t)$ is linear in $t$, then $r'(t)$ is a constant, so $r(t)$ cannot be written in the form $h(r'(t))$.
On the other hand, $g(x)$ doesn't have to be a 1-1 map.
The main advantage of this formalism is to express very conveniently the rule for derivating a composition function. In physics, one tends to give a name to the value of a function at a given point, rather than to the function itself. For example, using your functions $f$ and $g$, one would write :
$$y=g(x)$$
$$z=f(g(x))$$
$$\frac{dz}{dx}=\frac{dz}{dy}\times\frac{dy}{dx}$$
In mathematics books, one rarely sees such notations. It is more usual to write :
$$(f\circ g)'=(f'\circ g)\times g'$$
The main problem with the former notation is that, if $z$ is a function of two variables $y$ and $u$, the derivation rule above is wrong ! That is probably the reason for the symbol $\partial$ used with functions of several variables and partial derivatives instead of $d$. One should then write, for example :
$$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial y}\times\frac{\partial y}{\partial x}+\frac{\partial z}{\partial u}\times\frac{\partial u}{\partial x}$$
