Can $\mathrm{d}x$ be thought of as a derivative and differentiation or it's just a small change in $x$ and nothing more?

Question

The $\mathrm{d}x$ appears on integrals. I saw conflicting views regarding it. People sometimes write it does have a connection to differentiation and derivatives. Does it or does it not?

Answer 1

$dx$ means $\Delta{x}$, or an infinitesimal change in $x$. The $\frac{d}{dx}$ in derivatives is simply a mathematical operator that you apply to a function; it comes from $\frac{\Delta{y}}{\Delta{x}}$ = $\frac{\Delta}{\Delta x}y=\frac{d}{dx}y$.

Answer 2

It all depends on the context in which $dx$ appears.

If $x$ is a function of $t$, then $$dx=x'(t)dt$$ involves derivative.

If $dx$ appears in an integral, it simply means integrate with respect to $x$.

If it appears as a denominator such as $\frac {df}{dx}$ it mean differentiate with respect to $x$

If it is standing by itself it means an increment in $x$