Why Derivative of e^x is e^x

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In calculus we all eventually learn about the the mysterious exponential function that is its own derivative. If we choose Euler’s number, e,  as the base for the exponential function, it turns out that this particular function is its own derivative.  The exponential function with base e is the only exponential function whose tangent line has a slope of one at the origin. What follows is an explanation of why this is the case.

One can tell by graphing that the limit you see above will approach 1 for some number between 2 and 3. It turns out that the number, e=2.718 , is the value that produces a limit of one which is also the slope of the tangent line to the exponential curve at x=0. This is not really a rigorous proof, but provides strong numerical evidence that the derivative of the   exponential function with e as its base is itself.

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