Partial derivative chain rule?

  • Thread starter tade
  • Start date
  • #1
552
18

Main Question or Discussion Point

Is [tex]\frac{∂y}{∂x}×\frac{∂x}{∂z}=-\frac{∂y}{∂z}[/tex]?
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
No, the chain rule does not involve such a minus sign.
Why are you asking?
 
  • #3
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
It is true that IF the 3 variables x,y,z are related together in a sufficiently nice condition G(x,y,z)=0, THEN, we may solve for, in a region about a solution point, one of the variables in terms of the other two, i.e, functions X(y,z), Y(x,z), Z(x,y) exist, so that within that region we have that G(X(y,z),y,z)=G(x,Y(x,z),z)=G(x,y,Z(x,y)=0 identically.

In these cases, it is true that we have the counter-intuitive result:
[tex]\frac{\partial{Y}}{\partial{x}}\frac{\partial{X}}{\partial{z}}\frac{\partial{Z}}{\partial{y}}=-1[/tex]
 
  • #4
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
To give a simple example.
Consider the function G(x,y,z)=x+y+z

Now, the condition G(x,y,z)=0 gives rise to the equation x+y+z=0
We may now form three separate function definitions:
X(y,z)=-y-z
Y(x,z)=-x-z
Z(x,y)=-x-y

We have now that:
G(X(y,z),y,z)=-y-z+y+z=0, i.e, the condition G=0 is satisfied IDENTICALLY, for all choices of y and z.
Similarly with the other two substitutions.

We see that in this case, that we have:
[tex]\frac{\partial{Y}}{\partial{x}}=\frac{\partial{X}}{\partial{z}}=\frac{\partial{Z}}{\partial{y}}=-1[/tex]
and therefore,
[tex]\frac{\partial{Y}}{\partial{x}}\frac{\partial{X}}{\partial{z}}\frac{\partial{Z}}{\partial{y}}-1[/tex]
 
  • #5
552
18
No, the chain rule does not involve such a minus sign.
Why are you asking?
Try using the simple example z = x + y

Isn't there a minus sign?
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
41,833
955
In that specific case, the equation is true but it is NOT "the chain rule". Your initial post implied that you were offering this as a general formula derived from the chain rule.
 
  • #7
552
18
In that specific case, the equation is true but it is NOT "the chain rule". Your initial post implied that you were offering this as a general formula derived from the chain rule.
Is there a general formula for partial derivatives or is it a collection of several formulas based on different conditions?

Anyway, consider a function w ( x, y, z ). We can then derive z ( x, y, w ).

In this case, is
[tex]\frac{∂w}{∂z}×\frac{∂z}{∂x}=-\frac{∂w}{∂x}?[/tex]
 
  • #8
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
"w ( x, y, z ). We can then derive z ( x, y, w )."
This is meaningless.
 

Related Threads on Partial derivative chain rule?

  • Last Post
Replies
6
Views
16K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
673
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
1K
Replies
5
Views
2K
Replies
6
Views
4K
Replies
1
Views
806
  • Last Post
Replies
13
Views
3K
  • Last Post
Replies
5
Views
1K
Top