Why is the derivative of e^x equal to e^x?

[Jonathan48]

I'm trying to understand why the derivative of e^x is itself. I get the proof using the limit definition, but it still feels like a strange, almost magical result. I keep wondering if there's a more intuitive, visual way to see why this specific function's rate of change is exactly its current value at every single point.

[HannahL]

I used to stare at a graph of e to the x and try to feel it rather than prove it. When the chart climbs the slope seems to chase the height itself and that makes the idea of derivative feel almost obvious. It helps to think of growth as a percentage so the number and the slope look tied together.

[Aaron_J]

I tried a tiny step experiment in a notebook. I looked at how much the height changes when x goes up by a tiny bit and saw that the change is almost the current height times that tiny step. So the slope ends up looking like the height because the current height is the rate and the value at that exact moment is the thing being measured.

[Chloe88]

I changed nothing and still felt puzzled for a while. I told myself this is not magic but a property of the way the function is defined and the frame of reference. A quick image I use is a growing bank account that earns interest continuously so the extra money keeps increasing in proportion to what you already have.

[George.R]

Maybe the real problem is that I am chasing a special coincidence and not the point. Is it really the case that this is the unique curve whose instantaneous rate matches its height or am I just expecting too much from intuition?