take a look at the limit for e my friend, lim(x->inf) (1 + 1/x)^x, now take a look at this limit, lim(x->inf) xth root of e, this would equal 1 right? yes, but would i be wrong in saying that it equals 1.000000...001? by looking at the limit for e, this makes sense. that is the same as lim(x->inf) e^(1/x), ok, with this we have 1.0...1, now with e^(2/x) we have 1.0..1 * 1.0..1 and with e^(3/x) we have 1.0..1 * 1.0...1 * 1.0...1. by increasing the power by the number that approaches 0, we increase the result by itsself multiplied by the same number. which means when there is a change in x that approaches 0, the change in y will be the change in x multiplied by the current value which means that e^x is its own derivative.