Is the Derivative w.r.t. a Function Different from the Classic Chain Rule?

In summary, the conversation is about taking derivative w.r.t. a function instead of an independent variable. The person asking the question is wondering if the equation df/dz = df/dx * dx/dz + df/dy * dy/dz is correct, given that z is a function and not an independent variable. However, it is later determined that this equation is not meaningful and the conversation turns to discussing the relationship between functions and derivatives. The proof for this relationship involves the chain rule and the inverse function theorem.
  • #1
ledol83
12
0
Hi, i have a question on taking derivative w.r.t. to a function (instead of an independent variable). Actually i saw an excellent post on this same forum but that one was about a single variable.

My question is: f is function of x and y, and z is some other function also dependent on x and y, so is the following correct?

df/dz=df/dx*dx/dz+df/dy*dy/dz

it differs from the classic chain rule in the sense that z is actually a function (not an independent var), so i am not sure about this.

I appreciate so much for any comment!
 
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  • #2
First, go back to how the derivative wrt to a function is defined. (functional derivative)
Second, be more precise about your specific question.
As I understood, f is a function of x and y: f(x,y)
therefore, if you confirm that, I would say the df/dz = 0 .
 
  • #3
Hi actually i have f(x,y) and z(x,y) and was just wondering if this is true:

df/dz=df/dx*dx/dz+df/dy*dy/dz

thanks a lot!
 
  • #4
No this cannot be true, since this has no meaning.
Tell us what you think the meaning of df/dz would be, maybe then we can help.
 
  • #5
i have realized that what i posed was not meaningful. i am now thinking over my problem again.. thanks!
 
  • #6
how could you have a derivative of a function with respect to another function that the first function is not a function of :rofl::rofl:
 
  • #7
If f(x) is a function of x and g(x) is a function of x, you can surely write f as a function of g.

In particular,
[tex]\frac{df}{dg}= \frac{df}{dx}\frac{dx}{dg}= \frac{\frac{df}{dx}}{\frac{dg}{dx}}[/tex]
 
  • #8
HallsofIvy said:
If f(x) is a function of x and g(x) is a function of x, you can surely write f as a function of g.

In particular,
[tex]\frac{df}{dg}= \frac{df}{dx}\frac{dx}{dg}= \frac{\frac{df}{dx}}{\frac{dg}{dx}}[/tex]

not that I'm doubting you personally but i don't see where this comes from; proof?
 
  • #9
"not that I'm doubting you personally but i don't see where this comes from; proof?"

The first point is to know when we are talking about f1=f(g) or f2=f(x),
The second point is about the class of functions considered,
Otherwise, this is trivial (assuming dg is smooth):

f'(g) = f(g+dg)/dg = f(g(x)+dg(x))/dg(x) = f(g(x) + g'(x) dx) / (g'(x) dx) = f'(x)/g'(x)
 
  • #10
ice109 said:
not that I'm doubting you personally but i don't see where this comes from; proof?

It is a consequence of the chain rule and the inverse function theorem. Maybe you should do some reading too instead of making fun of peoples questions. Better try to help or keep out.
 
Last edited:

1. What does it mean to take the derivative with respect to a function?

Taking the derivative with respect to a function means finding the rate of change of that function with respect to its independent variable. It is a mathematical operation that involves calculating the slope of a tangent line to the function's graph at a specific point.

2. How is the derivative with respect to a function different from the derivative with respect to a variable?

The derivative with respect to a function involves finding the rate of change of a function with respect to its independent variable, while the derivative with respect to a variable involves finding the rate of change of one variable with respect to another variable. In other words, the former is a function of the latter.

3. What is the chain rule and how is it used to find the derivative with respect to a function?

The chain rule is a calculus rule that allows us to find the derivative of a composite function. In other words, if a function is composed of two or more functions, the chain rule tells us how to find the derivative of the composite function by using the derivatives of its component functions.

4. Can the derivative with respect to a function be negative?

Yes, the derivative with respect to a function can be negative. This means that the function is decreasing at that specific point. The sign of the derivative indicates the direction of change of the function at that point - a positive derivative indicates an increasing function, while a negative derivative indicates a decreasing function.

5. Why is the derivative with respect to a function important?

The derivative with respect to a function is important because it allows us to analyze the behavior of a function and understand its rate of change at a specific point. It is used in many fields, such as physics, economics, and engineering, to model and predict the behavior of various systems.

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