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.
$\endgroup$ 32 Answers
$\begingroup$$$\frac{\ d}{\ dx}e^y$$
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 $u=e^y$.
$\endgroup$ $\begingroup$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:
$$(e^{y(x)})' = y'(x)e^{y(x)} = \frac{dy}{dx}e^y$$
$\endgroup$ 2
