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Introduction to Differential Equations - At A Glance:

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

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