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The derivative of e^x is e^x.

The derivative of a function is the rate at which the function is changing at a particular point. In the case of e^x, the rate of change is equal to the function itself. This means that the derivative of e^x is simply e^x.

To prove this, we can use the definition of the derivative:

f'(x) = lim(h->0) [f(x+h) - f(x)]/h

Let f(x) = e^x. Then:

f'(x) = lim(h->0) [e^(x+h) - e^x]/h

Using the laws of exponents, we can simplify this expression:

f'(x) = lim(h->0) [e^x * e^h - e^x]/h

f'(x) = lim(h->0) [e^x * (e^h - 1)]/h

Now we can use L'Hopital's rule to evaluate the limit:

f'(x) = lim(h->0) [e^x * e^h]/1

f'(x) = e^x * lim(h->0) e^h

Since lim(h->0) e^h = 1, we have:

f'(x) = e^x

Therefore, the derivative of e^x is e^x.

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