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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 $0$

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

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    $\begingroup$ 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. $\endgroup$ – amd Aug 31 '18 at 0:54
  • $\begingroup$ 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. $\endgroup$ – Ixion Aug 31 '18 at 0:57
  • $\begingroup$ You have a Samsung Galaxy S8. $\endgroup$ – amsmath Aug 31 '18 at 1:00
  • $\begingroup$ @amsmath a8 not s8 but what of it? Lol $\endgroup$ – William Aug 31 '18 at 2:32
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If $x=x(y)$ is a function of $y$, then

$$ \frac{d}{dy} e^x = x' e^x = \frac{dx}{dy} \cdot e^x $$

Otherwise it is zero as the calculator says.

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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 $y$.

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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.

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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 $0$

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