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$f(x,y)$ and $g(x,y)$ are differentiable functions.

How do I find an expression of partial derivative of $f$ with respect to $g$ while holding $x$ constant?

Is it just $\frac{df}{dy} \times \frac{dy}{dg}$?

(it is not a duplicate of another question, which concerns the derivative of $f(u,v)$ with respect to $u(x,y)$ or $v(x,y)$, which is explained by the chain rule).

My attempt enter image description here

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    Possible duplicate of [Differentiating with respect to a function](http://math.stackexchange.com/questions/237512/differentiating-with-respect-to-a-function) – 2017-01-05
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    Why do you want to differentiate a function with respect to another function? – 2017-01-05
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    @Rumplestillskin No idea, just an idea my professor asked me to think about. – 2017-01-05
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    Select two differentiable functions for both $f(x,y)$ and $g(x,y)$ and take the derivative of one with the other... what happens? Remove the arbitrariness and compute an actual example. – 2017-01-05
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    Furthermore, if it is something your professor asked you to think about. Stop and actually think about what you are doing when you take a derivative. – 2017-01-05
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    @Rumplestillskin The professor asked to keep the problem as arbitrary as possible. However, I've added my own attempt, but I'm not sure is it true or not (and I have a test soon so I don't have time to check with my professor). – 2017-01-05
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    Good job! All done and dusted! – 2017-01-05
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    Possible duplicate: https://math.stackexchange.com/questions/291376/differentiate-with-respect-to-a-function – 2017-12-07

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