First of all I want to clarify that I posted this question on many forums and Q&A websites so the chances of getting an answer will be increased. So don't be surprised if you saw my post somewhere else.

Now let's get started:

When it comes to definitions, I will be very strict. Most textbooks tend to define differential of a function/variable in a way like this:

--------------------------------------------------------------------------------Let $ \displaystyle f(x)$ be a differentiable function. By assuming that changes in $ \displaystyle x$ are small, with a good approximation we can say:

$ \displaystyle \Delta f(x)\approx {f}'(x)\Delta x$Where $ \displaystyle \Delta f(x)$ is the changes in the value of function. Now if we consider that changes in $ \displaystyle f(x)$ are small enough then we define differential of $ \displaystyle f(x)$ as follows:

$ \displaystyle \mathrm{d}f(x):= {f}'(x)\mathrm{d} x$Where $ \displaystyle \mathrm{d} f(x)$ is the differential of $ \displaystyle f(x)$ and $ \displaystyle \mathrm{d} x$ is the differential of $ \displaystyle x$.

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What bothers me is this definition is completely circular. I mean we are defining differential by differential itself. Although some say that here $ \displaystyle \mathrm{d} x$ is another object independent of the meaning of differential but as we proceed it seems that's not the case:

First of all we define differential as $ \displaystyle \mathrm{d} f(x)=f'(x)\mathrm{d} x$ then we deceive ourselves that $ \displaystyle \mathrm{d} x$ is nothing but another representation of $ \displaystyle \Delta x$ and then without clarifying the reason, we indeed treat $ \displaystyle \mathrm{d} x$ as the differential of the variable $ \displaystyle x$ and then we write the derivative of $ \displaystyle f(x)$ as the ratio of $ \displaystyle \mathrm{d} f(x)$ to $ \displaystyle \mathrm{d} x$. So we literally (and also by stealthily screwing ourselves) defined "Differential" by another differential and it is circular.

Secondly (at least I think) it could be possible to define differential without having any knowledge of the notion of derivative. So we can define "Derivative" and "Differential" independently and then deduce that the relation $ \displaystyle f'{(x)}=\frac{\mathrm{d} f(x)}{\mathrm{d} x}$ is just a natural result of their definitions (using possibly the notion of limits) and is not related to the definition itself.

Though I know many don't accept the concept of differential quotient($ \displaystyle \frac{\mathrm{d} f(x)}{\mathrm{d} x}$) and treat this notation merely as a derivative operator($ \displaystyle \frac{\mathrm{d} }{\mathrm{d} x}$) acting on the function($ \displaystyle f(x)$) but I think that it should be true that a "Derivative" could be represented as a "Differential quotient" for many reasons. For example think of how we represent derivatives with the ratio of differentials to show how chain rule works by cancelling out identical differentials. Or how we broke a differential into another differential in the $ \displaystyle u$-substitution method to solve integrals. And it's especially obvious when we want to solve differential equations where we freely take $ \displaystyle \mathrm{d} x$ and $ \displaystyle \mathrm{d} y$ from any side of a differential equation and move it to any other side to make a term in the form of $ \displaystyle \frac{\mathrm{d} y}{\mathrm{d} x}$, then we call that term "Derivative of $ \displaystyle y$". It seems we are actually treating differentials as something like algebraic expressions.

I know the relation $ \displaystyle \mathrm{d} f(x)=f'(x)\mathrm{d} x$ always works and it will always give us a way to calculate differentials. But I (as an strictly axiomaticist person) couldn't accept it as a definition of Differential.

So my question is:

Can we define "Differential" more precisely and rigorously?

Thank you in advance.

P.S. I prefer the answer to be in the context of "Calculus" or "Analysis" rather than the "Theory of Differential forms". And again I don't want a circular definition. I think it is possible to define "Differential" with the use of "Limits" in some way(though it's just a feeling).