Regarding the Use of dy/dx

Abusing notation. We do it a lot in math.

Indeed $\frac{dy}{dx}$ is Leibniz notation for the derivative of $y$ with respect to $x$. Let us then write $y=f(x)$. Doing the problem "correctly," when we have a separable equation, we end up with $$F(y)\frac{dy}{dx}=G(x)$$ or rather $$F(f(x))\frac{df}{dx}(x)=G(x)$$. So then we integrate both sides with respect to $x$ and get $$\int F(f(x))\frac{df}{dx}(x) dx=\int G(x) dx$$. At this point make the u-substitution $y=f(x)$. We obtain $$\int F(y)dy = \int G(x) dx$$. Knowing the theory, this is why we go from $$F(y)\frac{dy}{dx}=G(x)$$ to $$F(y) dy = G(x) dx$$ and integrate both sides to get $$\int F(y) dy = \int G(x) dx$$. Note that a similar reason goes into explaining our u-substitution step $u=f(x)$ and $du=f^{\prime}(x) dx$