Prove the formula for the derivative of e^x.

The derivative of e^x is e^x.

To prove the formula for the derivative of e^x, we start with 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 properties 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 formula for the derivative of e^x is e^x.