# What is the derivative of $e^x$ wrt $y$?

Back in high school, I was taught

$$\dfrac{d}{dy} e^x = e^x \dfrac{dx}{dy}$$

Then why do I see people on the internet saying it's $0$. Even the derivative calculator says it's $0$

I thought that was suppose to be partial differentiation's job to treat other variables as constants?

• You can’t fully evaluate ${d\over dy}e^x$ without specifying the relationship between $x$ and $y$. If there is none, then $dx/dy=0$ and your chain-rule expansion agrees with the calculator. – amd Aug 31 '18 at 0:54
• Well, if $x$ does not depend by $y$ then $\frac{d}{dy}e^x=0$, but if $x$ is a function of $y$, the chain rule can give a different result. – Ixion Aug 31 '18 at 0:57
• You have a Samsung Galaxy S8. – amsmath Aug 31 '18 at 1:00
• @amsmath a8 not s8 but what of it? Lol – William Aug 31 '18 at 2:32

If $x=x(y)$ is a function of $y$, then
$$\frac{d}{dy} e^x = x' e^x = \frac{dx}{dy} \cdot e^x$$
This is just a problem in dependence of the variables. If $x=x(y)$, then your statement holds by chain rule. The problem with the calculator is that it is assuming that x is a "constant" in terms of $y$.
In a strict sense, you are correct. But you have had to specify $y$ was a function of $x$. You could write it as $y[x]$. If you don't indicate a dependence is possible, it assumes $\frac{dy}{dx}=0$ because it has to simplify as much as it can. If it didn't make any simplifying assumptions, it wouldn't be able to do much.
Chain rule holds only for composite function.if you gave $x(y)$ then,you could apply chain rule.but,as the question is given,there $x$ is only a variable.It doesn't depend on $y$.so it doesn't have any change relative to y.hence the result is $0$