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Once upon a time, 
long, long time ago in a cave somewhere, 

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there was some cave person who first 
started studying numbers. 

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And then, the cave person had an idea, 
the idea is functions. 

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Instead of just studying numbers, this 
cave person is going to study numbers 

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depending on other numbers, 
the relationships between numbers. 

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There's another way to think about 
functions. 

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A different metaphor is that functions 
[NOISE] eat numbers and after they're 

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done processing them, they spit out 
[NOISE] some other number. 

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Functions eat numbers and spit out 
numbers, but the food chain keeps going. 

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The derivative is an operator, and what 
that means, is that it's like a function 

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for functions. 
The derivative eats a function, and after 

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it's done with it, it spits out a new 
function, the derivative. 

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First, we studied numbers, 
then we studied functions, which are 

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numbers that depend on numbers or things 
that you do to numbers, and now, the 

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derivative. 
The derivative eats something which 

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itself eats something and spits out 
numbers, 

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and then, the derivative spits out a new 
thing that eats numbers and spits out 

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numbers. 
So the derivative takes a function and 

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gives you a new function and this d/dx 
notation is so powerful that you can 

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start to talk about the derivative even 
before you've applied it to any function. 

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Look at this. 
This is some sort of equation, but it's 

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really a nonsense equation. 
The second derivative, minus one, what 

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does it even mean? Equals the derivative 
minus one. 

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What am I doing here, multiplying? 
Who knows? 

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The derivative of plus one, this sort of 
nonsense looks similar to something very 

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reasonable, 
that x^2-1=(x-1)(x+1). 

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There's actually some sense to this. 
So let's try to make sense of the 

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right-hand side after I apply it to a 
function. 

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So we don't even really know what this 
means, but the notation is so powerful 

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that it's just going to lead us forward. 
So I'm going to copy down (d/dx-1) in 

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parentheses and I had to figure out how 
I'm going to apply this thing to f. 

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I'm going to act like this distributes. 
I'll write d/dx of f plus 1*f. 

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Now, I can keep on going. 
All right? This, if you kind of imagine, 

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might distribute over this. 
What would that, what would that mean? 

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I'm going to d/dx this whole thing, so 
d/dx of d/dx of f plus f minus one times 

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this, so minus d/dx of f plus f. 
And now, I can keep calculating. 

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The derivative of the derivative of f 
plus f, well, the derivative of a sum is 

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the sum of the derivative, so that makes 
sense. 

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That's the derivative of the derivative 
of f plus the derivative of f minus the 

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derivative of f minus f, subtracting, 
summing f, so it's subtracting f. 

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I have good news. I've got a plus 
derivative here and a minus derivative of 

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f there, 
so what I'm left with is d/dx d/dx of f 

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which I could write as the second 
derivative of f minus f. 

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I could also write this as the second 
derivative minus one applied to f and 

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that's exactly what's on the other side 
of this equation. 

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At this point, this is all a sort of 
cheating, 

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so if this doesn't speak to you don't 
worry. 

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[MUSIC] The upshot though is that 
differentiation is an operation to apply 

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to functions and it's possible to reason 
precisely about differentiation in the 

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abstract just as an operation. 
[MUSIC]