Partial derivative with fixed variable

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Homework Help Overview

The discussion revolves around the calculation of a partial derivative in the context of a statistical mechanics problem, specifically involving the functions y = xz5 and x = zg, where g is a constant. Participants are exploring the implications of holding variables constant while differentiating.

Discussion Character

  • Conceptual clarification, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the concept of partial derivatives, particularly the challenge of holding one variable constant while differentiating with respect to another. There is uncertainty about how to treat the variables x and z in this context. Some suggest using a tree diagram to visualize the relationships and derivatives, while others question the relevance of such a diagram.

Discussion Status

The conversation is ongoing, with participants raising questions about the implications of treating g as a constant and the effects of varying g or z. There is recognition of a potential issue regarding the independence of x when z is held constant, prompting further exploration of the relationships between the variables.

Contextual Notes

Participants note the constraints of the problem, particularly the fixed nature of g and the implications this has on the differentiation process. There is a discussion about the assumptions underlying the problem setup and the potential for varying g, which complicates the interpretation of the partial derivative.

steve233
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Homework Statement



Given y = xz5 and x = zg (where g is some constant) find :

(∂y / ∂x)z

Homework Equations


The Attempt at a Solution



I understand the concept of a partial derivative, but I've never seen one such that there is a variable held fixed, or one where ∂x is not changing independently. This is a statistical mechanics problem. Any tips on how to do this?

My solution is basically treat z as a constant but that would be the same as taking the partial without holding z constant. I'm not sure what to do about x either. Just need an example of or instruction on how to do this.
Thanks.
 
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Draw a tree diagram (whatever they are called) first.
 
I'm not looking for a probability, so how would the tree diagram help?
 
The tree diagram flyingpig is talking about has nothing to do with probability. He is talking about taking f as "root" and drawing branches for "derivative with respect to x" and "derivative with respect to z", etc. in order to use the chain rule for partial derivatives.

However, in this very simple example, I don't think I would use the chain rule. From x= zg, it follows that z= x/g. Then y= xz^5= x(x^5/g^5)= x^6/g^5. diferentiate that.
 
And keep in mind that g is a constant...
 
Okay so basically I want to get rid of the term that is constant by using other variables?

Lets say for example instead of z being the constant in the question (the subscript) I choose g now to be the subscript. How would this change the partial derivative (Assuming g and z can both vary somehow)?

Also, the derivative is quite simple:

(∂y / ∂x)z = 6x5 / g5
 
steve233 said:

Homework Statement



Given y = xz5 and x = zg (where g is some constant) find :

(∂y / ∂x)z

Homework Equations


The Attempt at a Solution



I understand the concept of a partial derivative, but I've never seen one such that there is a variable held fixed, or one where ∂x is not changing independently. This is a statistical mechanics problem. Any tips on how to do this?

My solution is basically treat z as a constant but that would be the same as taking the partial without holding z constant. I'm not sure what to do about x either. Just need an example of or instruction on how to do this.
Thanks.

I think there may be a serious problem here: (dy/dx)_z means: take the partial of y wrt x, holding z constant. However, since x=zg and g is constant, when we hold z constant we cannot vary x at all!

RGV
 
Ah, I see the issue.
What if g can vary? What then?
 
Well, then g isn't a constant, which is at odds with what you said in the first post.
 

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