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
ELI5…
There are only two ways for comparing numbers. Subtraction: a – b = ? And quotient: a/b = ?
Now suppose you’re in a lab, you have to test some apparatus. You set a lever on position x0 and get a reading y0. To see what happens you nudge the lever to x0 + Δx (Δx being some small increment or decrement) and get a new reading y1. Wanting to compare numbers (see supra) you form:
Δy = y1 – y0 (the effect of your nudge)
k = Δy/Δx (the ratio of effect to nudge – subtraction is meaningless there)
If the apparatus is reasonably stable you now know that *in the vicinity of x0* you have:
y ~ y0 + k . Δx
Δy ~ k . Δx (equivalently)
Now the idea of differential calculus is to make the nudge infinitesimal, Δx ~ 0 (but Δx ≠ 0). You then get *exact* results (not only approximations) once you get rid of the infinitesimals. Let’s take an example, behavior of x^3 around x0 = 2:
y0 = x0^3 = 2^3 = 8
y = (x0 + Δx)^3 = x0^3 + 3.x0^2. Δx + 3.x0. Δx^2 + Δx^3 (binomial formula)
y = y0 + 3.2^3.Δx + Δx^2.( … )
y = 8 + 24.Δx + terms infinitesimal even when compared to Δx
Finally:
Δy = y – 8 = 24.Δx + yet smaller infinitesimals ~ 24.Δx
Δy/Δx ~ 24 (true for any infinitesimal Δx)
But this being a property of the polynomial x^3 at the number x0 = 2 (no infinitesimals there, it’s just the slope of the tangent) it has to be exactly 24:
Δy/Δx = 24
Or, changing to standard notation, dy/dx = 24.
Admittedly that’s a hairy part when you start dabbling in [Nonstandard analysis](https://en.wikipedia.org/wiki/Nonstandard_analysis). The Great Old Ones like Leibnitz used infinitesimals somewhat nonchalantly – and were justly criticized for that, but baron Cauchy later formalized the calculus with the familiar ε – δ reasoning on limits (but still used infinitesimals!).
I hope all this blather explains the origins of the notation. And it works. Let’s take the chain rule (f and g differentiable, of course);
z = g( f( x ) )
f( x0 + dx) ~ f( x0 ) + k.dx = y0 + k.dx
g( y0 + dy) ~ g ( y0 ) + k’.dy ~ z0 + k’.k.dx
z0 + dz ~ z0 + k’.k.dx
dz ~ k’.k.dx
dz/dx = k’.k = g'( y0 ). f'( x0 ) = g’ ( f( x0 ) ). f’ ( x0 )
It boils down to an elementary property of linear functions: if c = k’.b and b = k.a than c = k’.k.a. And that’s what we do, approximating g and f by their linear parts (tangents).
Of course the notation has taken a life of its own (e. g. a basis for vector fields on a manifold would be notated ∂/∂u_i like an operator because vector fields can be, and are, considered as operators) but I hope I’ve managed to convey the original meaning. (Hey, it’s ELI5!)
Note that when reasoning with infinitesimals there is no need for limits the Cauchy way: *an infinitesimal stands for all possible ways to go to zero*. If interested Keisler’s [Elementary Calculus: An Infinitesimal Approach](https://www.math.wisc.edu/~keisler/calc.html) is free to download.
Now I need a beer… 😉
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