Partial derivative of a function with respect to another function

Asked Jan 05 '17 at 11:34

Active Jan 05 '17 at 13:12

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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

edited Jan 05 '17 at 13:12

asked Jan 05 '17 at 11:34

The First StyleBender

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Possible duplicate of Differentiating with respect to a function – Jan 05 '17 at 11:40
Why do you want to differentiate a function with respect to another function? – Rumplestillskin Jan 05 '17 at 13:04
@Rumplestillskin No idea, just an idea my professor asked me to think about. – The First StyleBender Jan 05 '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 05 '17 at 13:07
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. – Rumplestillskin Jan 05 '17 at 13:08
@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). – The First StyleBender Jan 05 '17 at 13:13
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Good job! All done and dusted! – Rumplestillskin Jan 05 '17 at 13:18
Possible duplicate: https://math.stackexchange.com/questions/291376/differentiate-with-respect-to-a-function – Felipe G. Nievinski Dec 07 '17 at 04:26

Partial derivative of a function with respect to another function

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