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]