I recently came across a question that asked for the derivative of $e^x$ with respect to $y$. I answered $\frac{d}{dy}e^x$ but the answer was $e^x\frac{dx}{dy}$. How is that the answer? I am confused.
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$\begingroup$If $x$ is a function of $y$ then the given answer follows by the chain rule:
$$\frac {\text d}{\text dy} \left(e^x\right) = e^x \cdot \frac {\text d}{\text dy}(x)$$
$\endgroup$ $\begingroup$The question you have been asked is a bad question, if you weren't given more information.
If $e^x$ is considered a function of two (independent) variables $x$ and $y$, then "derivative" probably means "partial derivative", and $\frac{\partial}{\partial y}e^x=0$.
If there is some relation between $x$ and $y$, then the chain rule applies.
$$\frac{d}{dy}(e^x)=\frac{d}{dx}(e^x)\cdot\frac{dx}{dy}=e^x\frac{dx}{dy}$$
And your answer $\frac{d}{dy}e^x$ is technically correct, but it's basically restating the question.
$\endgroup$ $\begingroup$Informally, the question understood it on the way, that how the $y$ coordinate changes of the curve $(x;e^x)$ for the infinitesimally small changes of the $x$ coordinate. This is the common interpretation of the derivate, but it is not the only one. You have probably got this question in differential calculus. Your answer was correct, but missed exactly this little important detail.
$\endgroup$ $\begingroup$I feel the question needs to be clearer. It needs to be mentioned what exactly is $y$. Like one of the comment says, $y$ may not be a dependent variable. In that case, $\frac{\partial{e^{x}}}{\partial{y}}$ equates to zero. However, in case $x$ is a function of $y$ then, $\frac{de^{x}}{dy}$ can be written as $$\frac{de^{x}}{dy}×\frac{dx}{dx}$$ $$=\frac{de^{x}}{dx}×\frac{dx}{dy}$$ This gives the answer as, $$e^{x}×\frac{dx}{dy}$$
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