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

[dipole]

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 $f = f(x,a)$

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 $df/da = 0$ seems like the only thing that makes sense, but does it actually?

For example, if $f(x,a) = ax + a$, then should $df/da = 0$ or $df/da = x + 1$ ?

 

[Mark44]

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.

 

[AlephZero]

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".

 

[oli4]

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...

 

[chiro]

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?