When to write "$dx$" in differentiation

I'm taking differential equations right now, and the lack of fundamental knowledge in calculus is kicking my butt.

In class, my professor has done several implicit differentiations. I realize that when taking the derivative with respect to "$x$," I have to write "$\frac{dy}{dx}$" whenever I differentiate "$y$" and nothing when it's "$x$" since it will be "$\frac{dx}{dx}$."

However, today, he differentiated the equation "$y=ux$" to get "$d y= u\, d x + x\, d u$." I'm not sure why he has decided to write $d y$ and $d x$ separately like that. I don't think he did implicit differentiation. Can someone explain what he did? Thank you very much.

EDIT: To provide more context, the professor was working on changing a homogeneous differential equation to a separable differential equation. He stated that in $f(x,y)$, all $y$ must be substituted with $ux$. Therefore, $f(x,ux)$. This prompted him to find the derivative of the equation as well, which he wrote as "$dy = u\,dx + x\,du$"

Sometimes we write $$ d y= u d x + x d u $$ and understand it to mean $$ \frac{d y}{dt}= u \frac{d x}{dt} + x \frac{d u}{dt} $$ where $x, y$ and $u$ are all functions of some variable $t$, perhaps $t$ is yet to be determined...

It is simply the product rule and chain rule applied:

$$\require{cancel} y = u(x) x \implies \frac{dy}{dx} = u \frac{dx}{dx} + \frac{du}{dx}x \implies dy = u\, dx \frac{\cancel{dx}}{\cancel{dx}} + x\, \cancel{dx} \frac{du}{\cancel{dx}}.$$