I Like Abstract Stuff; Why Should I Care?

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 that may eat and/or spit out multiple numbers. The function

f (u, v) = + v, u – v>

eats two numbers and spits out a vector. 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.