It’s pretty common to hear that dy/dx isn’t a fraction, but if that’s the case then why do we treat as such in a differential equation or an integral? For example, if dy/dx = f(x), then how can we just write it as dy= f(x)dx as though it’s a fraction?
In: Mathematics
Simply put, “dy/dx” is an instruction to take a very specific limit. It’s not an instruction to calculate 2 numbers called dy and dx and then take the ratio. That’s why we say not to treat them like fractions.
The reason you can do manipulations in differential equations where you write dy/dx=f(x) as dy=f(x)dx is because you eventually write the latter with an integral sign. So what you’re really saying is that dy/dx=f(x) if and only if integral dy = integral f(x) dx.
Now, recall that by the fundamental theorem of calculus integration is the inverse of differentiation. Also recall that multiplication is the inverse of division. So when you adopt the differentials-as-fractions notation, all thats really happening is that you’re saying is that we aim to undo the derivative, derivatives kinda look like fractions, so we’ll write the inverse of the derivative in the same way that we would write the inverse of a fraction, and it works out nicely.
Tl:dr: the fundamental theorem of calculus provides convenient notation
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