Interpreting Derivatives with Respect to a Constant: Do They Actually Change?

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Discussion Overview

The discussion revolves around the interpretation of derivatives with respect to a constant in the context of thermodynamics and mathematical functions. Participants explore the implications of differentiating a function that includes a constant parameter and the conceptual understanding of such derivatives.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions how to interpret the derivative of a function f = f(x,a) with respect to a constant a, suggesting that if a cannot change, then df/da should equal 0.
  • Another participant argues that taking a derivative with respect to a constant does not make sense and claims it is undefined.
  • A different participant encourages providing a specific thermodynamics problem to clarify the confusion regarding differentiation with respect to a constant.
  • Another viewpoint suggests that if a appears in the function definition, it should be treated as a variable or parameter, implying that it can influence the function's behavior even if it is constant in a specific context.
  • One participant introduces a related example involving a constant in a two-dimensional system, prompting further thought about derivatives in that context.

Areas of Agreement / Disagreement

Participants express differing views on whether it is meaningful to take derivatives with respect to constants, with some asserting it is undefined while others propose a more flexible interpretation. No consensus is reached on the interpretation of such derivatives.

Contextual Notes

Participants highlight the ambiguity surrounding the definition of constants and parameters in mathematical functions, as well as the potential variations in physical constants in different contexts.

dipole
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This is coming up a lot in some of my thermo HW, so I'm a little confused about whether or not I'm thinking correctly.

Suppose I have a function [itex]f = f(x,a)[/itex]

where a is some constant. If I take the derivative of f wrt to a, what do I get?

The derivative tells you the change in the function due to some small change in a quantity, so if I want to know how f changes with respect a, how does one interpet this?

a can't change, so it seems that f can't change wrt to a, so [itex]df/da = 0[/itex] seems like the only thing that makes sense, but does it actually?

For example, if [itex]f(x,a) = ax + a[/itex], then should [itex]df/da = 0[/itex] or [itex]df/da = x + 1[/itex] ?
 
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It doesn't make any sense to take a derivative wrt a constant. So IOW, taking the derivative of f wrt to a is undefined.
 
As Mark44 said, the question you asked doesn't mean anything, so it's hard to know what you are really having problems with.

Can you post a particular thermo problem where you think you need to differentiate wrt a constant? That might get a more helpful answer than "this doesn't mean anything".
 
Don't worry about 'a' being a constant, if it appears in the function definition, then it is a 'variable' as any other.
Think of it as a parameter if it helps
For instance, suppose f(x, a) is a function that gives you for x being the height, how long will a mass takes to hit the ground. 'a' would be the constant 'g'
Obviously, this 'constant' isn't so constant, it will vary slightly on earth, and a lot more if you don't stay on earth, you could become interested in asking yourself, just how much do different values of g change the value of f for some fixed x...
 
Hey dipole and welcome to the forums.

One thing that you might want to think about is the situation y = c for a constant c. Now let's say our system is two dimensional (x,y). What is dy/dx? How about dx/dy?
 

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