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I've seen other posts about finding a derivative with respect to another function, but I didn't understand how it would work when the functions have more than one variable.

I would like a general explanation about how to find $\frac{df(x,y)}{dg(x,y)}$, but take the following functions to illustrate it:

Given the functions

$$f(x,y) = 3x + 5y$$ $$g(x,y) = 2x + y$$

is it possible to find $\frac{df}{dg}$?

Thiago Rangel
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  • Take a look at this: https://math.stackexchange.com/questions/954073/derivative-of-a-function-with-respect-to-another-function – nullgeppetto Sep 13 '17 at 21:34
  • Yeah, I've seen it, but it's different, because there are two variables (x and y) in my example. I couldn't figure out how to solve it. – Thiago Rangel Sep 13 '17 at 21:36
  • @ThiagoRangel See also: https://math.stackexchange.com/questions/307411/derivative-of-fx-y-with-respect-to-another-function-of-two-variables-kx-y – nullgeppetto Sep 13 '17 at 21:37
  • One issue is that your second function has two independent variables, so you really need to ask about partial derivatives. HOWEVER, you might think along the lines of directional derivatives. If you had a vector pointing in the same direction as the line represented by g(x, y), then you could calculated the directional derivative of f in this direction. The only other thing I could think of is to take a linear combination of the partial derivatives of f, and use the Chain Rule somehow. You would need to check for consistency though. BTW notice that f(x, y) = 1.5 g(x, y) + 3.5 y. Interesting... – KenM Feb 01 '21 at 21:11

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