Question

why is the differential of e^y = e^y ?

Answer 1

if my question isnt clear please let me know..

Answer 2

Here's a quick proof, using the chain rule:
\[
u=e^x\]We know:\[
\frac{d}{dx}\ln(e^x)=\frac{d}{dx}x=1
\]So, we begin:\[
\frac{d}{dx}\ln(e^x)=\frac{d}{du}\ln(u)\frac{d}{dx}e^x=\frac{1}{u}\cdot\frac{d}{dx}e^x=\frac{1}{e^x}\cdot\frac{d}{dx}e^x=\]Which implies:\[
\frac{d}{dx}e^x=e^x
\]

Answer 3

I meant:
\[
\frac{1}{e^x}\cdot\frac{d}{dx}e^x=1
\](By our first derivation, on the top)

Answer 4

e is such a nice number ...look at the series of expansion of e^x .. and differentiate it.