Understanding Differentials (dx, dq, etc) in Physics Problems

[FS98]

When solving physics problems sometimes we have to use differentials like dx or dq. I don’t quite understand how to use these.

I understand that the limit as change in x approaches 0 of change in y over change in x is represented by dy/dx, where dy and dx are sometimes said to be small changes in x and y.

What I don’t understand is why something like dx can be sometimes treated as change in x. What’s the reasoning behind this?

 

[fresh_42]

Differentiation means a linear approximation, a tangent or a tangent space. This is a vector space and its basis vectors are sometimes written as $dx, dy, \ldots$ It is the linear approximation in the direction $x$, resp, $y, \ldots$ So in this sense it is the marginal linearized change in that direction.
Maybe this can answer a few questions:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/#toggle-id-1
There are many different views possible for differentiation: the point, the resulting function, the resulting tangent space, the linear approximation, the process itself, and all are only a differentiation. In the end it's always a linear approximation at some given point in some given direction(s). $dx$ is a short cut for this direction.
With a capital letter, $D_p f$ is a short cut for $\left. \dfrac{d}{d \vec{x} } \right|_p \, f(\vec{x}) = \sum \left. \dfrac{\partial f}{\partial x_i} \right|_{p_i} \, d x_i$ with the basis vectors $d x_i$ of the tangent space.