>>14104123The derivative is a function. The output represents the slope of the derived function at a given input.
The integral is a function. The output represents the sum of all outputs of the integrated function before the given input.
I always hated area under the curve. Integration is about having all this baggage and that's driving the current value of output. Derivation is about moving the future value as close to the present value as possible and then predicting the future.
Two big takeaways:
Because the function is continuous, you can do this. The past, the present, and the immediate future all relate. Any little break in continuity, even way way back, messes this up.
You can do this for other values than time. Gibbs did this with space. If you know a continuous function of a spatial region, you can zoom in as far as you want. It's tough to find a reason why something is not continuous, and this led to Maxwell finding electrical properties of empty space.
