Calculus was originally formulated in terms of infinitesimals. Hundreds of years later, a second formulation was found in terms of limits. There were originally some doubts about whether the version using infinitesimals was logically OK, but these doubts were cleared up by Robinson and others ca. 1970.

The Leibniz notation $\int f(x) dx$ was invented in the earlier period, so in this notation, $dx$ a notation for an infinitesimal. You can imagine a Riemann sum with the narrow rectangles having an infinitesimal width $dx$. The thing inside the integral sign is the width of one such rectangle: its height $f(x)$ multiplied by its width $dx$. Since multiplication is commutative, we have literally $f(x)dx=dxf(x)$.

Some math teachers today still tell their students that the $dx$ is just punctuation, or only functionsas a statement of what variable is being integrated with respect to. This could be because they are afraid their students will be confused by talk about infinitesimals, or because the teachers themselves don't know that any concerns about the logical issues with infinitesimals have been cleared up.

My question, is this notation recognized by others, in particular mathematicians as I myself have only seen it used in various papers and texts of topics in mathematical physics. Also, who used it first[...]

The notation was first used by Leibniz, with the factors written in whichever order was convenient. It is universally recognized, with the factors in either order, by people who understand the above historical and mathematical facts.

Calculus was originally formulated in terms of infinitesimals. Hundreds of years later, a second formulation was found in terms of limits. There were originally some doubts about whether the version using infinitesimals was logically OK, but these doubts were cleared up by Robinson and others ca. 1961.

The Leibniz notation $\int f(x) dx$ was invented in the earlier period, so in this notation, $dx$ a notation for an infinitesimal. You can imagine a Riemann sum with the narrow rectangles having an infinitesimal width $dx$. The thing inside the integral sign is the width of one such rectangle: its height $f(x)$ multiplied by its width $dx$. Since multiplication is commutative, we have literally $f(x)dx=dxf(x)$.

Some math teachers today still tell their students that the $dx$ is just punctuation, or only functionsas a statement of what variable is being integrated with respect to. This could be because they are afraid their students will be confused by talk about infinitesimals, or because the teachers themselves don't know that any concerns about the logical issues with infinitesimals have been cleared up.

My question, is this notation recognized by others, in particular mathematicians as I myself have only seen it used in various papers and texts of topics in mathematical physics. Also, who used it first[...]

The notation was first used by Leibniz, with the factors written in whichever order was convenient. It is universally recognized, with the factors in either order, by people who understand the above historical and mathematical facts.