notation - Difference between $Delta$, $d$ and $delta$

I have read the thread regarding 'the difference between the operators $\delta$ and $d$', but it does not answer my question.

I am confused about the notation for change in Physics. In Mathematics, $\delta$ and $\Delta$ essentially refer to the same thing, i.e., change. This means that $\Delta x = x_1 - x_2 = \delta x$. The difference between $\delta$ and $d$ is also clear and distinct in differential calculus. We know that $\frac{dy}{dx}$ is always an operator and not a fraction, whereas $\frac{\delta y}{\delta x}$ is an infinitesimal change.

In Physics, however, the distinction is not as clear. Can anyone offer a clearer picture?