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"e is the unique number such that e^x is equal to its derivative."

I haven't ever really understood why, but have figured it would pop up some time later. Unfortunately, it still hasn't popped up, so I've decided to ask. How is e^x the derivate of itself?

I was mucking around with numbers a bit and came up with a couple of things:

Asumming e^x is its derivative:

[tex]y = e^{x^{2}+2}[/tex]

Then:

[tex]\frac{dy}{dx} = e^{x^{2}+2}[/tex]

However (using the chain rule):

[tex]\frac{dy}{dx} = 2xe^{x^{2}+2}[/tex]

Which isn't the same thing...

Also, while I'm on the topic, is there a way to calculate NP derivatives such as: [tex]2^{x}[/tex]?

I am sorry if this seems basic, or is commonly asked, but I just can't seem to figure it out.