calculus - Implicit derivative of $e^y$

I am confused about this problem of finding the derivative of $e^y$ when differentiating with respect to $x$. The whole problem is to differentiate $y = x \, e^y$ with respect to $x$ but I get stuck on $\frac{d}{dx}(e^y)$.

I use the chain rule and end up with
$(e^y)(y)(\frac{dy}{dx})$, derivative of the outside times inside times derivative of the inside, but when I look up online to check my answer it seems that $\frac{d}{dx}(e^y) = (e^y)(\frac{dy}{dx})$. I'm confused where my extra $y$ went?

Any help would be greatly appreciated.

Answer

$$\frac{\ d}{\ dx}e^y$$

First take the derivative like you "normally would":

$$e^y$$

Then take the derivative of the stuff substituted "inside", the stuff where an $x$ would usually be:

$$\frac{\ d}{\ dx} y=\frac{\ dy}{\ dx}$$

Multiply them together.

$$e^y • \frac{dy}{dx}$$

Summarized mathemetically,

$$\frac{du}{dx}=\frac{du}{dy}•\frac{dy}{dx}$$

Where here $u=e^y$.