Implicit derivative of $e^y$

Question

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 1

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

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 $u=e^y$.

Answer 2

You can see it more clearly if you write it down this way:

$$\frac{d}{dx}e^{y}=\frac{d}{dx}e^{y(x)} = (e^{y(x)})'$$

So just apply the chain rule:

$$(e^{y(x)})' = y'(x)e^{y(x)} = \frac{dy}{dx}e^y$$