Bdtyktl

The derivative of a function is often defined as $f'$ and $f'(x)$. So which one is it? $f'(x)$ is the output of the function $f'$, so why do I see people using $f'$ and $f'(x)$ interchangeably to refer to the derivative of a function?

By definition a function is a triple $(f,D,C)$, which is very often denoted by $f colon D to C$, where $C,D$ are two sets and $f$ associates to each element of $D$ one and only one element of $C$.

So when it is clear what $C$ and $D$ are, or in cases where it is not possible or not necessary to write them down, you just write $f$. The expression $f(x)$ denotes the element in $C$ which $x in D$ is mapped to. So $f$ is a function, $f(x)$ is an element of $C$, two completely different things.

$f$ denotes the function and $f(x)$ the output of the function when evaluated at $x$.

This convention does not differ for the derivative.

I would read $f'(x)$ as "the function $f'$ applied to the element $x$ of the domain". This gives us a a new element in the range. Meanwhile I read $f'$ as a relation, it tells us which elements are mapped to which other elements. The prime just tells us that it is relation to some other function $f$ in a very specific way (derivation).

Example:
Our 'input' set is ${1,2,3 }$ our output set is ${A,B,C,D}$
$$f={(1,C),(2,A),(3,D) }$$
So we now know that $f(1)=C$ and $f(2)=A$. Notice that the element $B$ is not reached and this function is not surjective.

What you should take from this finite example is that a function is a rule that tells us which elements are in a way "paired", while $f(x)$ tells us about a specific pair. However sometimes people just represent the function like this by saying:

For arbitrary $x$ (so in our example $1$,$2$ or $3$), $f(x)$ is given by $dots$
This is indeed another representation of the same idea, but mathematicians ofter prefer the "relation" idea.

The derivative of the function $f$ is $f'$. People usually make the mistake of saying that it is $f'(x)$, just like they talk about, say, the function $sin(x)$, when, in fact, they should be talking about the $sin$ function.