Relationship between differential and derivative for non-linear functions

Question

Let us assume that we have a function: $$ y=f\left(x\right) $$ If $f\left(x\right)$ is linear and equal to $ax$, then we obtain: $$ \frac{dy}{dx}=a $$ where $a$ is the linear coefficient. However, in this case, if we take the differential of $y,$ we get $$ dy=a\times dx $$ which is re-written as: $$ \frac{dy}{dx}=a $$ In this case, we can think of $\frac{dy}{dx}$ as a ratio of the change in $y$ to the change in $x.$ However, I have read elsewhere that $\frac{d}{dx}$ is best thought of as an operator that operates on $y$. Is the distinction in a non-linear setting? In other words, assume we have: $$ y=f\left(x\right) $$ where $f(x)$ is non-linear. The differential of $y$ gives: $$ dy=f'\left(x\right)dx $$ Can the derivative be derived in this setting equivalently?