notation question on f and f(x)

Question

i have always thought that $f(x)$ denotes a function of $x$ and that $f$ represents this function in a wider sense, as the set of ordered pairs that relate elements between two sets.

When my book says suppose $f$ can be differentiated then the derivative denoted $f'$ is a function, i cant understand it.

it would make sense if it were $f(x)$ and $f'(x)$

Is it a problem with the way i'm thinking about the notation, thanks.

Answer 1

Formally $f$ is the function itself and $f(x)$ is the value of the function at the point $x$. But many times $f$ and $f(x)$ are just treated as the same thing. Now, if your function is differentiable at every point in the interval then you can define a function $f'$, it makes sense.