I thought I knew that we use differentials just because they made calculus intuitive and there was math rigor behind all of these. But when I saw this 3B1B video about the topic this problem came out.
For example, deriving implicitly: $$ x^2+y^2 = 5 $$ we get $$ 2x\mathrm{d}x +2y\mathrm{d}y = 0 $$ $$ 2xdx+2y\frac{dy}{dx}dx=0 $$ and then to find the slope of the graph we solve for dy/dx $$ \frac{dy}{dx}=-\frac{x}{y} $$
Does't that imply that $dx \neq 0$ ?
And how would it be in the general case whe differentiating implicitly