• Support PF! Buy your school textbooks, materials and every day products Here!

Partial derivatives

  • Thread starter ehrenfest
  • Start date
1,996
1
[SOLVED] partial derivatives

1. Homework Statement
Can the partial derivative of a function depend depend on the form it is in?

Say, z = f(x,y), and y=g(x,w). If I take

[tex]\frac{\partial z}{\partial y} [/tex]

then I get

[tex]\frac{\partial f(x,y)}{\partial y}[/tex]

which is not necessarily 0. But [itex]\frac{\partial z}{\partial y} [/itex] is also equal to

[tex]\frac{\partial f(x,g(x,w))}{\partial y}[/tex]

which is identically 0. This is DRIVING ME OUT OF MY MIND.

Also, say we have z = f(x,y) = x^2+y^2+y and we also know that x=y. Then z also equals g(x,y) = x^2+y^2+x.

Thus, we get

[tex]2 y +1 = \frac{\partial f(x,y)}{\partial y} = \frac{\partial z}{\partial y} = \frac{\partial g(x,y)}{\partial y} = 2y[/tex]

which is absurd. What is wrong with my logic?
All of these examples come from thermodynamics.
2. Homework Equations



3. The Attempt at a Solution
 
Last edited:

Answers and Replies

Dick
Science Advisor
Homework Helper
26,258
618
Of course the partial derivative depends on the form of the function. When you write a partial derivative you are implicitly assuming that some combination of variables is held constant. When you juggle the form around like that you are changing what you are thinking of as 'constant'. Don't do that. Use the chain rule for partial derivatives and everything will take care of itself.
 
979
1
To expound on what Dick just said --- make sure you say which variables are being held constant, *explicitly*.
 
1,996
1
Of course the partial derivative depends on the form of the function. When you write a partial derivative you are implicitly assuming that some combination of variables is held constant.
So, you are saying that partial derivatives do not make sense unless you hold enough variables constant to make the partial derivative unambiguous? That is, whenever I write down I partial derivative, I should always make sure that I have specified enough variables SO THAT THE PARTIAL DERIVATIVE IS INDEPENDENT OF THE FORM OF THE FUNCTION, right?

How do you know when you have specified enough variables to make the partial derivative unambiguous?
 
Last edited:
Dick
Science Advisor
Homework Helper
26,258
618
Uh, when you've specified enough that the function is only a function of one unfixed variable.
 
1,996
1
Uh, when you've specified enough that the function is only a function of one unfixed variable.
What about the first two questions in my last post?
 
HallsofIvy
Science Advisor
Homework Helper
41,732
893
The answer to both of those questions is "yes".
 

Related Threads for: Partial derivatives

  • Last Post
Replies
4
Views
717
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
811
Replies
4
Views
835
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
0
Views
2K
Top