Confused by how to derive the derivative of $f(\boldsymbol{x})=g(\boldsymbol{y})$ I was watching an online tutorial and saw this derivation.

It seems the the author took the derivative with respect to y on left side and to x on right side. I thought dx should always be in the denominator and should on both side of the equation. Is it partial derivative? Or maybe my misunderstanding of the notation? 
Could anyone explain how this works? FYI the link of the tutorial is https://www.youtube.com/watch?v=aXBFKKh54Es&list=PLwJRxp3blEvZyQBTTOMFRP_TDaSdly3gU&index=98, the differentials was taken at around 2'20"
Much appreciated! Happy New Year.

It's curious that you should call this a "derivation" because this is more accurate than saying that the derivative in the usual sense was taken. The construction used was the exterior derivative, or differential. This satisfies
$$df(x,y)=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$
for a function depending only on $x$ and/or $y$, so if $f(x,y)=\ln x$, then
$$df(x,y)=\frac{1}{x}dx+0dy=\frac{dx}{x}$$
and if $g(x,y)=\ln y$ then
$$dg(x,y)=0dx+\frac{1}{y}dy=\frac{dy}{y}$$
The reason it is called a derivation is for any functions $p,q$ we have
$$d(pq)=qdp+pdq$$
