The definition of the derivative of $y = f(x)$ at $x$, given that the limit exists is
$$\frac{dy}{dx} = \lim_{x \to 0} \frac{y}{x}$$
where $f(x + x) - f(x) = y$.
However, I wonder can we write $\frac{dy}{dx} = \lim_{y \to 0} \frac{y}{x}$ instead?
My idea would be that we can't do that. Because, $x$ is the independent variable. And $y$, as the dependent variable, changes with $x$. That means, as $x → 0$, we automatically have $y → 0$. However, we can't do it the other way round. Do you agree with my idea? I want to here your opinions. Thank you!
You are almost right. The point is that $δy$ is a function depending on $δx$. This becomes clearer if one does not use the these two names and writes $$ f'(x) = \lim_{h→0}\frac{f(x + h) - f(x)}{h} $$ instead.