Question

Prove that d/dx e^x = e^x without assuming it

Answer 1

Consider $$\log e^x=x$$hence $$\frac{d}{dx}\log e^x=\frac{d}{dx}x\\\frac1{e^x}\frac{d}{dx}e^x=1\\\frac{d}{dx}e^x=e^x\\Q.E.D.$$