Once upon a time, long, long time ago in a cave somewhere,
there was some cave person who first started studying numbers.
And then, the cave person had an idea, the idea is functions.
Instead of just studying numbers, this cave person is going to study numbers
depending on other numbers, the relationships between numbers.
There's another way to think about functions.
A different metaphor is that functions [NOISE] eat numbers and after they're
done processing them, they spit out [NOISE] some other number.
Functions eat numbers and spit out numbers, but the food chain keeps going.
The derivative is an operator, and what that means, is that it's like a function
for functions. The derivative eats a function, and after
it's done with it, it spits out a new function, the derivative.
First, we studied numbers, then we studied functions, which are
numbers that depend on numbers or things that you do to numbers, and now, the
derivative. The derivative eats something which
itself eats something and spits out numbers,
and then, the derivative spits out a new thing that eats numbers and spits out
numbers. So the derivative takes a function and
gives you a new function and this d/dx notation is so powerful that you can
start to talk about the derivative even before you've applied it to any function.
Look at this. This is some sort of equation, but it's
really a nonsense equation. The second derivative, minus one, what
does it even mean? Equals the derivative minus one.
What am I doing here, multiplying? Who knows?
The derivative of plus one, this sort of nonsense looks similar to something very
reasonable, that x^2-1=(x-1)(x+1).
There's actually some sense to this. So let's try to make sense of the
right-hand side after I apply it to a function.
So we don't even really know what this means, but the notation is so powerful
that it's just going to lead us forward. So I'm going to copy down (d/dx-1) in
parentheses and I had to figure out how I'm going to apply this thing to f.
I'm going to act like this distributes. I'll write d/dx of f plus 1*f.
Now, I can keep on going. All right? This, if you kind of imagine,
might distribute over this. What would that, what would that mean?
I'm going to d/dx this whole thing, so d/dx of d/dx of f plus f minus one times
this, so minus d/dx of f plus f. And now, I can keep calculating.
The derivative of the derivative of f plus f, well, the derivative of a sum is
the sum of the derivative, so that makes sense.
That's the derivative of the derivative of f plus the derivative of f minus the
derivative of f minus f, subtracting, summing f, so it's subtracting f.
I have good news. I've got a plus derivative here and a minus derivative of
f there, so what I'm left with is d/dx d/dx of f
which I could write as the second derivative of f minus f.
I could also write this as the second derivative minus one applied to f and
that's exactly what's on the other side of this equation.
At this point, this is all a sort of cheating,
so if this doesn't speak to you don't worry.
[MUSIC] The upshot though is that differentiation is an operation to apply
to functions and it's possible to reason precisely about differentiation in the
abstract just as an operation. [MUSIC]