# Partial derivative chain rule?

## Main Question or Discussion Point

Is $$\frac{∂y}{∂x}×\frac{∂x}{∂z}=-\frac{∂y}{∂z}$$?

## Answers and Replies

CompuChip
Homework Helper
No, the chain rule does not involve such a minus sign.
Why are you asking?

arildno
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Gold Member
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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:
$$\frac{\partial{Y}}{\partial{x}}\frac{\partial{X}}{\partial{z}}\frac{\partial{Z}}{\partial{y}}=-1$$

arildno
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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:
$$\frac{\partial{Y}}{\partial{x}}=\frac{\partial{X}}{\partial{z}}=\frac{\partial{Z}}{\partial{y}}=-1$$
and therefore,
$$\frac{\partial{Y}}{\partial{x}}\frac{\partial{X}}{\partial{z}}\frac{\partial{Z}}{\partial{y}}-1$$

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?

HallsofIvy
Homework Helper
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.

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
$$\frac{∂w}{∂z}×\frac{∂z}{∂x}=-\frac{∂w}{∂x}?$$

arildno
Homework Helper
Gold Member
Dearly Missed
"w ( x, y, z ). We can then derive z ( x, y, w )."
This is meaningless.