Hey, there's more!
First, why "d"? Well, "d" is for "difference". As in: as x changes from x_1 to x_2, the difference (x_2 - x_1) -- when it's very small.
But wait, there's more!
The commonly used symbols for finite difference like that is the Greek letter Delta: Δ
For a list of values x_1, x_2, x_3, x_4,.. we write Δx_i = (x_i - x_{i-1}). That is, Δx_i is the i'th change. (Side note: an airline had a marketing slogan Change is Delta, which some nerd must have been immensely proud of).
Ok, bear with me for a bit more!
The symbol we use for finite sums is Σ: we write
Σy_i = y_1 + y_2 + ... + y_n
Why Σ? That will have to wait a little.
Sums and differences cancel out:
ΣΔy_i = (y_1 - y_0) + (y_2 - y_1) + (y_3 - y_2) + ... + (y_n - y_{n-1}) = y_n - y_0
That is, summing up small succesive changes gives you total change. Simple?
Now apply this to the situation where the small changes in the quantity you are looking at are proportional to changes in another:
Δy_i = Δx_i * f(x_i)
Say, y is position, x is time; then f(x_i) is the speed at time x_i: as time increases a little, so does your position; the ratio of the changes is the speed. Δy_i is how much you moved from time x_{i-1} to time x_i, which is proportional to change in time
Δx_i.
Note that f(x_i) = Δy_i/Δx_i here (speed = change in position div. by change in time).
Now write:
y_n - y_0 = ΣΔy_i = Σf(x_i)Δx_i
Again, just summing up small changes to get the net chnage.
NOW, what you've been waiting for!
Imagine you took infinitely many measurements. The changes become infinitely small, and the sum becomes of infinitely many things.
We need new notation for this.
But let's keep it similar. Instead of using Greek letters, let's use the same letters... in Latin.
Δx becomes dx
ΣΔy becomes S dy
And, with some sloppy handwriting of the letter S, the net change equation becomes:
y_final - y_initial = ∫ f(x)dx
where
f(x) = dy/dx.
You now see that Σ and the sloppy S -- ∫ -- stand for Sum.
And that, my friend, is pretty much all there is to Calculus and its symbols, fundamentally[1].
[1]https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_calcu...