We're used to functions that eat numbers and spit out other numbers. The function
f (x) = x2
eats a number and spits out the square of that number. For example,
f (4) = 16.
There are also vector functions (link) that may eat and/or spit out multiple numbers. The function
f (u,v) = (u + v,u - v)
eats two numbers and spits out two other numbers. For example,
f (2,3) = (2 + 3,2 - 3) = (5,-1).
Now, for extra fun, there are functions that eat functions and spit out other functions! The act of taking a derivative is one example. We can think of as a function that eats a function and spits out the derivative of that function (which is itself a function):
For example, if we feed the function f (x) = x2 to the function we get
The function ate the function f (x) = x2 and spit out its derivative function, f '(x) = 2x