Is the derivative a function?

[Zermelo]

Hi guys, this is just a quick question that came to my mind that I would like to discuss. The question from the thread’s name could be easily misinterpreted, I’m not asking if the derivative of a function is a function itself (f’ is a function, which is obvious), but I’m wondering if the “action” of taking a derivative is a function?
I think that the derivative satisfies all the conditions for being a function on the set of differentiable functions on R, D(R). Let’s denote d as the derivative function. [MATH]d: D(R) -> F(R)[/MATH], where F(R) is the set of all real functions. And d(f) = lim df/dx. Every differentialble function has a derivative, and a function cant have 2 derivatives, thus the derivative is a function itself.
This seems pretty clear to me, but I have never heard of anybody addressing the derivative as a function, for example, everyone says that the derivative is linear, but no one says it’s a linear function. Maybe I’m wrong and it’s not a function, and maybe people don’t talk about it as a function just not to cause confusion?

 

[lex]

Not wanting to promote rival sites(!) but there is a similar post here, which may be of interest:

The Derivative of a Linear Operator

Why is the derivative (d/dx) thought of as a linear operator instead of a function of functions? if we take the derivative of some function f(x) (d/dx(f(x))), then we get a new function f’(x). T...

math.stackexchange.com

 

[Zermelo]

Not wanting to promote rival sites(!) but there is a similar post here, which may be of interest:

The Derivative of a Linear Operator

Why is the derivative (d/dx) thought of as a linear operator instead of a function of functions? if we take the derivative of some function f(x) (d/dx(f(x))), then we get a new function f’(x). T...

math.stackexchange.com

Didn’t know there was rivalry xD
This is pretty much what I needed, thanks, it was just weird to me that I don’t hear the term “derivative function” ever

 

[lex]

Didn’t know there was rivalry xD
This is pretty much what I needed, thanks, it was just weird to me that I don’t hear the term “derivative function” ever

Yes, I suppose I only really heard it described as a 'differential operator'.

 

[Steven G]

I too have heard of it as a differential operator.
I consider the derivative to be a limit given how it is defined.

 

[Dr.Peterson]

Hi guys, this is just a quick question that came to my mind that I would like to discuss. The question from the thread’s name could be easily misinterpreted, I’m not asking if the derivative of a function is a function itself (f’ is a function, which is obvious), but I’m wondering if the “action” of taking a derivative is a function?
I think that the derivative satisfies all the conditions for being a function on the set of differentiable functions on R, D(R). Let’s denote d as the derivative function. [MATH]d: D(R) -> F(R)[/MATH], where F(R) is the set of all real functions. And d(f) = lim df/dx. Every differentialble function has a derivative, and a function cant have 2 derivatives, thus the derivative is a function itself.
This seems pretty clear to me, but I have never heard of anybody addressing the derivative as a function, for example, everyone says that the derivative is linear, but no one says it’s a linear function. Maybe I’m wrong and it’s not a function, and maybe people don’t talk about it as a function just not to cause confusion?

You're right. And you aren't alone, though I agree that we tend not to say it quite that way to avoid confusion.

See here: https://en.wikipedia.org/wiki/Derivative#The_derivative_as_a_function

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by D, then D(f) is the function f′. Since D(f) is a function, it can be evaluated at a point a. By the definition of the derivative function, D(f)(a) = f′(a).​

Now, what is the inverse of this function?

 

[Zermelo]

[MATH][/MATH]

You're right. And you aren't alone, though I agree that we tend not to say it quite that way to avoid confusion.

See here: https://en.wikipedia.org/wiki/Derivative#The_derivative_as_a_function

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by D, then D(f) is the function f′. Since D(f) is a function, it can be evaluated at a point a. By the definition of the derivative function, D(f)(a) = f′(a).​

Now, what is the inverse of this function?

The derivative is obviously injective, and if we restrict the codomain to I(R), where I(R) is the set of all real functions that have an antiderivative, then the inverse would be the integral from 0 to x of f(t) with respect to t ?
I’m not really familiar with topology, but I know that there are some spaces in which a limit can be defined, and then I guess the derivative could also be defined in those spaces. It would be really cool if D(R) was one of these spaces, then we could get the derivative of the derivative?

EDIT:
After thinking about it, I think that every metric space can have a limit defined in it. Thus, D(R) is a metric space (we can define a metric using integrals), and then we can find a limit that represents the derivative of a derivative?

EDIT 2
The derivative of the derivative would be [MATH] \lim _{h -> 0} \frac{d(f(x)+h) - d(f(x))}{h}[/MATH], but d(f+h) = d(f) + d(h) = d(f) (I guees h is still a real constant in this weird case?) Then the derivative of the derivative = 0?

 

[HallsofIvy]

The question "Is the derivative a function" is ambiguous. The operation of taking the derivative is an "operation" not function. The result of that operation, i.e. "the derivative of f(x)", IS a function.

 

[Zermelo]

The question "Is the derivative a function" is ambiguous. The operation of taking the derivative is an "operation" not function. The result of that operation, i.e. "the derivative of f(x)", IS a function.

I clarified the ambiguity in the thread’s description.
Look at the other answers, we concluded that it is indeed a function on D(R). And I don’t know how you define operations, but I learned at my college that a (binary) operation on set A is a function f: AxA -> A, so even in that case the derivative is a function, although I don’t think the derivative is an operation, because it only takes single functions as inputs.