Partial derivative with fixed variable

[steve233]

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.

 

[flyingpig]

Draw a tree diagram (whatever they are called) first.

 

[steve233]

I'm not looking for a probability, so how would the tree diagram help?

 

[HallsofIvy]

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.

 

[Mark44]

And keep in mind that g is a constant...

 

[steve233]

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

 

[Ray Vickson]

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

 

[steve233]

Ah, I see the issue.
What if g can vary? What then?

 

[Mark44]

Well, then g isn't a constant, which is at odds with what you said in the first post.