# Partial Derivative Problem

1. Jul 5, 2015

### slr77

1. The problem statement, all variables and given/known data
Define f(x,y) = x+2y, w = x+y. What is ∂f / ∂w?

2. Relevant equations

3. The attempt at a solution
f = w+y so:

∂f/∂w = ∂(w+y)/∂w = ∂w/∂w + ∂y/∂w = 1 + ∂y/∂w. But I'm really not sure if this is right and if it right so far, I can't figure out what ∂y/∂w should be...

2. Jul 5, 2015

### RyanH42

This is right.

Now partial derivative means the derivative of given parameter.The other parameters will be assumed constant.Cause they are not changing respet to that parameter.I mean If I set two parameter a and b and make a fuction using them (It can be anything).Lets call it f(a,b)=a+b then ∂f/∂a means derivative of function respet to a not b.So we will assume b is constant and ∂f/∂a=1 so.Do the same thing.

3. Jul 5, 2015

### Staff: Mentor

@RyanH42: the problem here: which other variable is supposed to be constant? Why should it be y (as you seem to suggest), not x, or x-y? Those would lead to different answers.

4. Jul 5, 2015

### slr77

But w depends implicitly on y so can I really take y as constant? If I get y in terms w (y = w-x) and continue this way (so 1 + ∂y/∂w = 1 + ∂(w-x)/∂w). I just get an endless chain of 1+1+1+1+1... That's why I think what I'm doing is not right.

5. Jul 5, 2015

### RyanH42

I dont know but he answer might be this ∂f/∂w=∂f/∂y.∂y/∂w+∂f/∂x.∂x/∂w look this rule.

6. Jul 5, 2015

### slr77

I think this is the chain rule but we haven't learned that yet. I'll read ahead and come back to this and make sense of it but apparently there should be a way to do this without directly making use of the chain rule.

7. Jul 5, 2015

### RyanH42

I have an idea ∂f/∂w=1 + ∂y/∂w now ∂f/∂w=1 + 1/∂w/∂y then ∂f/∂w=2 Its a trick but I dont know its ture or not.I know it was stupid idea

Last edited: Jul 5, 2015
8. Jul 5, 2015

### verty

This problem can't be solved, it is underspecified.

So ∂y/∂w + 1/∂w/∂y = 0? It wasn't actually a stupid idea, you just made an algebra mistake.

9. Jul 5, 2015

### slr77

Hmm, ok. I think I should have posted the full problem because I think it's more open ended than what my original post conveys:

I'm just treating w as the variable and going from there but maybe that's not the right definition? So is there some way to do this problem that makes sense?

10. Jul 5, 2015

### verty

I still maintain this is a flawed question and you should move on. It doesn't contain enough information to answer it.

11. Jul 5, 2015

### RyanH42

Is this true ? or You mean ∂w/∂w + 1/∂w/∂y ? I am confused

12. Jul 5, 2015

### RyanH42

If you use chain rule you get 2 again.I think answer is 2.
$∂f/∂w=∂f/∂y.∂y/∂w+∂f/∂x.∂x/∂w$
$∂f/∂y=2$
$∂y/∂w=1/2$
$∂f/∂x=1$
$∂x/∂w=1$
So answer is 2 I guess.

Whats your's idea ?

Last edited: Jul 5, 2015
13. Jul 5, 2015

### verty

$f(x,y) = x + 2y$
${∂f \over ∂w} = {∂x \over ∂w} + 2 {∂y \over ∂w}$

The problem happens because we don't know what ${∂x \over ∂w}$ and ${∂y \over ∂w}$ are, we don't have enough information to determine them. If this isn't clear, be sure to look again at partial derivatives and what they mean.

Last edited: Jul 5, 2015
14. Jul 5, 2015

### Staff: Mentor

w=x+y. Why should the two derivatives be different?

Also, the rule doesn't work like that with partial derivatives. You could introduce arbitrary new parameters (e. g. z=x) and add more and more terms that would change the result.

15. Jul 5, 2015

### RyanH42

Here my last idea then.The question ask's us for general solution and we do the general solution : ${∂f \over ∂w} = {∂x \over ∂w} + 2 {∂y \over ∂w}$ or
${∂f \over ∂w} = {∂f \over ∂x}{∂x \over ∂w} +{∂f \over ∂w}{∂y \over ∂w}$
Part b asks solve these equation with spesific parameters.I think there must be some difference between question a and b so I thought we can think x and y like numbers or actually constant parameters.So I mean f=x+2y is a constant cause x and y is contant so the answer is zero.

16. Jul 6, 2015

### RyanH42

Forgive me but Why you guys stop answering the question.Theres a problem and you are avoiding to answer.If one of you find the answer he can tell here cause I am curios.

17. Jul 6, 2015

### ChrisVer

The question (a) asks for the definition of the derivative of a general function f(x,y)...The definition of the derivatives is with limits I guess...even with or without limits, you can write your 2nd expression in p#15, and try to find the dx/dw, dy/dw.

(b) asks for the given function: f=x+2y

18. Jul 6, 2015

### Staff: Mentor

Maybe we are overthinking this. There is no explicit dependence on w in f...