Partial derivative chain rule?

In summary, the conversation discusses the relationship between partial derivatives and the chain rule. It is stated that the chain rule does not involve a minus sign and this is shown through a simple example. The question is asked if there is a general formula for partial derivatives, to which it is replied that there is not, but rather a collection of formulas based on different conditions. Another example is given where the function w (x, y, z) is considered and z (x, y, w) is derived. It is then asked if the equation \frac{∂w}{∂z}×\frac{∂z}{∂x}=-\frac{∂w}{∂x} is true in this case
  • #1
tade
702
24
Is [tex]\frac{∂y}{∂x}×\frac{∂x}{∂z}=-\frac{∂y}{∂z}[/tex]?
 
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  • #2
No, the chain rule does not involve such a minus sign.
Why are you asking?
 
  • #3
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
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
CompuChip said:
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
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
HallsofIvy said:
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
"w ( x, y, z ). We can then derive z ( x, y, w )."
This is meaningless.
 

1. What is the partial derivative chain rule?

The partial derivative chain rule is a mathematical rule used to find the derivative of a function with multiple variables. It is used when one variable is dependent on another variable, and both variables are functions of a third variable. It allows us to find the rate of change of a function with respect to one variable while holding the other variable constant.

2. How do you use the partial derivative chain rule?

To use the partial derivative chain rule, you must first identify the dependent and independent variables in the function. Then, take the derivative of the outer function with respect to the variable in question, treating all other variables as constants. Finally, multiply this derivative by the derivative of the inner function with respect to the same variable.

3. Why is the partial derivative chain rule important?

The partial derivative chain rule is important because it allows us to solve complex problems involving multiple variables, which are common in science and engineering. It also helps us understand the relationship between different variables in a function and how they affect its rate of change.

4. What is the difference between the partial derivative chain rule and the regular chain rule?

The partial derivative chain rule is used for functions with multiple variables, while the regular chain rule is used for functions with only one variable. The partial derivative chain rule also involves treating all other variables as constants, whereas the regular chain rule does not.

5. Can the partial derivative chain rule be applied to any function?

Yes, the partial derivative chain rule can be applied to any function with multiple variables as long as the variables are related to each other in a chain-like manner. It is a fundamental rule in multivariable calculus and is used in various fields of science and engineering.

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