Partial derivative with fixed variable

In summary, the conversation discusses finding the partial derivative (∂y/∂x)z given the equations y = xz5 and x = zg, where g is a constant. The concept of a partial derivative and the use of tree diagrams is mentioned, and it is noted that in this example, the chain rule may not be necessary. The final solution is given as (∂y/∂x)z = 6x^5/g^5, but a potential issue is raised if g is able to vary.
  • #1
steve233
20
0

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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  • #2
Draw a tree diagram (whatever they are called) first.
 
  • #3
I'm not looking for a probability, so how would the tree diagram help?
 
  • #4
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 [itex]y= xz^5= x(x^5/g^5)= x^6/g^5[/itex]. diferentiate that.
 
  • #6
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
 
  • #7
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
 
  • #8
Ah, I see the issue.
What if g can vary? What then?
 
  • #9
Well, then g isn't a constant, which is at odds with what you said in the first post.
 

1. What is a partial derivative with a fixed variable?

A partial derivative with a fixed variable is a mathematical concept that involves taking the derivative of a multivariable function with respect to one of its variables while holding the other variables constant. It allows us to measure the rate of change of a function in one direction while keeping all other variables fixed.

2. How is a partial derivative with a fixed variable calculated?

To calculate a partial derivative with a fixed variable, we follow the same rules as finding a regular derivative. We differentiate the function with respect to the variable of interest and treat all other variables as constants. This means that we can use the power rule, product rule, quotient rule, and chain rule, depending on the complexity of the function.

3. Why do we use partial derivatives with fixed variables?

We use partial derivatives with fixed variables to understand how a function changes in one specific direction while keeping all other variables fixed. This allows us to analyze the behavior of a function in a specific direction and is particularly useful in fields such as physics, economics, and engineering where variables are often interdependent.

4. What is the difference between a partial derivative and a total derivative?

A partial derivative measures how a function changes in one direction while keeping all other variables fixed. On the other hand, a total derivative measures how a function changes in all directions, taking into account the effects of changes in all variables. In other words, a total derivative includes the effects of both partial derivatives and is calculated using the chain rule.

5. Can a partial derivative with a fixed variable be negative?

Yes, a partial derivative with a fixed variable can be negative. This means that the function is decreasing in the direction of the variable of interest while keeping all other variables fixed. It is important to note that the sign of a partial derivative with a fixed variable can change depending on the values of the other variables in the function.

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