nonstandard analysis - Do normal arithmetics apply for infinitesimal variables?

In physics, we tend to use non-standard analysis, meaning we use the same arithmetics (division, multiplication, even vector arithmetics) on infinitesimal variables ($df, dx, dy, ...$) as on normal ones. This makes a lot of calculation (see below) intuitive. But we also "silently assume" certain propertities (like smoothness, integrableness, etc.) of most functions and variables intuitively, so we may pass some mathematical restrictions without noticing.

Some examples for 1D derivation (let $f = f(x), x = x(y)$ in the 1D, $f = f(x, x_2, x_3, ...), x = x(y, y_2, y_3,...)$ in the N-dimension case) are below.:

1. Changing derivation variable (chain rule):
   $$\frac{df}{dx} = \frac{df}{dx} \frac{dy}{dy} = \frac{df}{dy} \frac{dy}{dx}$$


2. 
   

   Changing integration variable:
   $$\int{f dx} = \int{f dx \frac{dy}{dy}} = \int{\left(f \frac{dx}{dy}\right)dy}$$

   
   


3. 
   

   "Reciprocal of dervation":
   $$\frac{1}{g} = \frac{df}{dx} \longrightarrow g = \frac{dx}{df}$$

   
   


4. 
   

   Solving separable differential equations, e.g.:
   $$ \frac{dg}{dx} = f(x) $$ $$dg = f(x)dx$$ $$g(x) = \int{dg} = \int{f(x)dx}$$

   
   

Most of these are things I know from standard calculus, and they are indeed intuitive, but I'm curious how correct this notation is. I wonder, can you threat infinitesimal variables always like regular ones, possibly in other, more complicated cases where non-standard methods are non-intuitive? What restrictions apply (e.g. smooth, derivable/integrable n times, real, etc.), and are these general or situation-dependent? If the latter, what is the biggest set of functions where this kind of infinitesimal arithmetics work?

Answer

The rule such computations generally rely upon is the transfer principle of Robinson's framework for analysis with infinitesimals. Of course, one needs to learn how to apply the principle correctly. As far as partial derivatives are concerned, an inexperienced student can make mistakes with those as much as a student of Robinson's framework.