A question on notation used in differential equations by George F Simmons

Question

In page number 57 of Differential Equations by George F Simmons a rather peculiar notation is used in the section on integrating factors, that is when he talks about a method of reducing to exact differential equation, it goes like this in equation (11) $$d(\dfrac{y}{x}) = \dfrac{xdy-ydx}{x^2}$$ What is it? It looks like the quotient rule but it doesnt really make sense to me.

Answer 1

Yes, this is the quotient rule, packaged as a total differential $$ dF =\partial_xF\,dx+\partial_yF\,dy $$ of the function $$ F(x,y)=\frac yx $$ to get $$ dF=-\frac{y}{x^2}\,dx +\frac{1}x\,dy=... $$