# Differential Equations

# 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*) = *x*^{2}

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*) = *x*^{2} to the function we get

The function ate the function *f* (*x*) = *x*^{2} and spit out its derivative function, *f* '(*x*) = 2*x*