Taking derivative of a differential equation

Question

When I was reading my lecture note examples, I came across some expression about a DE like this. Edit: $y$ is a function of $x$. $$ y'=y-x-1$$ And for $$f=y-x-1\to f'= \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} f $$ I have some experience with both multi-variable calculus and DE, but this second expression had me lost. What exactly happened there?

Answer 1

Combining first and second equation $$ y’ = y + x + 1 = f $$ you get $f=y’=dy/dx$, otherwise it is the total differential of $f$ divided by $dx$. $$ \frac{df}{dx} = \frac{\partial f}{\partial x} \frac{dx}{dx} + \frac{\partial f}{\partial y} \frac{dy}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f $$