Partial derivative of a function with respect to another function

$f(x,y)$ 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

• Possible duplicate of Differentiating with respect to a function – user371838 Jan 5 '17 at 11:40
• Why do you want to differentiate a function with respect to another function? – Rumplestillskin Jan 5 '17 at 13:04
• @Rumplestillskin No idea, just an idea my professor asked me to think about. – The First StyleBender Jan 5 '17 at 13:05
• 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. – Rumplestillskin Jan 5 '17 at 13:07
• Good job! All done and dusted! – Rumplestillskin Jan 5 '17 at 13:18