The difference between $\Delta x$, $\delta x$ and $dx$

$\Delta x$ is about a secant line, a line between two points representing the rate of change between those two points. That's a "differential" (between the two points).

$dx$ is about a tangent line to one point, representing an instantaneous rate of change. That makes it a "derivative."

$\delta x$ is about a tangent line to a partial derivative. That's a rate of change or derivative in one direction, holding a number of other directions constant.

$\Delta x$, is used when you are referring to "large" changes, e.g. the change from 5 to 9. $\partial x$ is used to denote partial derivative when you have a multivariate function (e.g. one with x,y,w, instead of just x alone). $dx$ is used to denote the derivative when you have a univariate function (when you just have x and there is no confusion).

There are several answers to similar/the same questions:

• Given $z=f(x,y)$, what's the difference between $\frac{dz}{dx}$ and $ \frac{\partial f}{\partial x}$?
• What is the difference between $d$ and $\partial$?

But the answer from Tom Au also puts it in a nutshell.