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 

Yes its valid we use this result in cases where we want to find an approximate value without using calculator. For eg say we want the value of $\sqrt {64.1} $ so we define $f (x)=\sqrt {x}$ then using your equation its $\sqrt {64.1} \approx \sqrt {64}+0.1 \frac {1}{2\sqrt {64}}=8+0.1/16=8.00625$
