Is $f(x + dx) -f(x) = f'(x) \,\mathrm dx$ a valid equation? $\def\d{\mathrm{d}}$We know that it is true that$$\lim_{\Delta x \to 0} \frac{f(x + \Delta x) -f(x)}{\Delta x} = \frac{f(x + \d x) -f(x)}{\d x} = f'(x),$$
where $\d x$ is define to be an infinitesimal.
Then we could rearrange the equation and say that$$f(x + \d x) -f(x) = f'(x) \,\d x.$$
Will this last equation be valid or correct?
Update. Do you agree that:
$$f'(x) \,\d x = \int_{x}^{x+\d x}f'(x)\,\d x.$$
Is this last equation valid or making sense? Does it even mean anything if you put a $dt$ in the limit? What I meant by valid is that would it be possible to apply it like in the context of the question  here 

In Questions involving these kind of tricks, Taylor series expansion is always useful and very powerful 
From Taylor Series Expansion of $f(x + dx)$ about $x=x$, 
\begin{equation}
f(x+dx)=f(x)+f^{'}(x)dx+f^{''}(x)(\frac{(dx)^{2}}{2!}) +f^{'''}(x)(\frac{(dx)^{3}}{3!}) + ....
\end{equation}
If $f(x)$ is defined and all derivatives are defined at $x=x$ then we can approximate above equation by neglecting higher order terms under following assumption that our step-size $dx$ tends to $0$. 
And we have following relation
\begin{equation}
f(x+dx)\approx f(x)+f^{'}(x)dx
\end{equation}
\begin{equation}
f(x+dx)- f(x) \approx f^{'}(x)dx
\end{equation}  
This is just an approximation though, not exact equal sign because you have neglected higher order terms. 
This limiting taylor series expansion has numerous applications especially in Computational-science and numerical methods for solving ordinary differential equations. 
(Sorry, I don't have good latex skills though :) )
Update: To answer your question in update, I don't see error in that equation but again it should have approximate sign and the derivative should be defined
