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

Question

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?

Answer 1

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.

Answer 2

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 $y$.

Answer 3

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.

Answer 4

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 $0$