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why is the derivative of e^x e^x
Look at the curve y equals e to the x. Where it is 2.7 tall, it climbs with slope 2.7. Where it is 7.4 tall, slope 7.4. Steepness equals height, everywhere. Why?
For any exponential a to the x, slope is change over h. Factor out a to the x and you get the height times a leftover that never mentions x.
That leftover is just the slope at x equals zero. Base two gives 0.693, base three gives 1.099. Somewhere between sits a base giving exactly one. That base is e, about 2.71828.
So for e the leftover is one, and the slope is e to the x times one: the height itself. Any constant multiple C times e to the x does the same.
Three curves, all starting at height one. Their starting slopes are 0.69, one, and 1.10, and each slope stays that number times the height. Only e makes it exactly one.
The derivative of e to the x is itself because e was chosen as the one base whose slope at zero is exactly one.