Usually, the first thing about e that one learns in school is the definition e=lim n→∞ (1+1/n)^n. Later on one is taught if f(x)=e^x then f'(x)=e^x. Most school textbooks on mathematics however lack proof for this, even though it is quite simple.
From the definintion of the derivative: f'(x)=(f(x+Δx)-f(x))/Δx when Δx→0
f'(x)=(e^(x+Δx)-e^x)/Δx
f'(x)=(e^x*e^Δx-e^x)/Δx
f'(x)=e^x*(e^Δx-1)/Δx
Now, all that remain is to show that (e^Δx-1)/Δx→1 as Δx→0
To do this one inserts e=(1+1/n)^n when n→∞.
f'(x)=e^x*((1+1/n)^n*Δx-1)/dx
As n→∞ and Δx→0, we can get rid of one of the limits by writing n=1/Δx. This makes the above equation simplify to
f'(x)=e^x*((1+Δx)^1 - 1)/Δx
f'(x)=e^x*Δx/Δx=e^x