Proofing the derivatives of e^x from the limit approach

Hetware, maybe it's lame question, but where did the last line come from? I mean, the thing

1=e^x d(e^x)/dx
 Recognitions: Homework Help ^ 1=x'=(log(e^x))'=(e^-x)(e^x)'
 Thanks, don't know how I could miss that :).