How can I explain the equation $y' = \frac {dy}{dx}=f(y)g(x)$ to someone who has only taken precalculus?
Context: a friend asked my to explain the first page of my PDE textbook. I found it amazingly challenging to describe exactly (or at all) the difference between each side of the equivalence.
I said: y' is just a symbol we use to denote the idea that a number y is changing in relation to something else. The next part shows that the "something else" is x. In fact, when x changes a bit and y changes a bit, the ratio between those two changes is going to be dy/dx.
I couldn't find a good way to describe the last part though.