[MUSIC] In the future, we're going to 
have a lot of very precise statements 
about the derivative. But before we get 
there, I want us to have some intuition 
as to what's going on. 
Let's take a look at just the S, I, G, N, 
the sign of the derivative. 
The thick green line of the plot is some 
random function, and the thin red line is 
its derivative. And note that when its 
derivative is positive, the function's 
increasing, and when the derivative is 
negative, the function's decreasing. 
We can try to explain what we're seeing 
here formally, where that calculation on 
paper. 
So let's suppose that the derivative is 
positive over a whole range of values. 
And, we also know something about how the 
derivative is related to the functions 
values. 
The function's output or x+h is close to 
the functions output of x plus how much 
the derivative tells us the output should 
change by, which is how the input changed 
by times the ratio change of output 
change to input change. 
Alright. Now let's suppose that x+h is a 
bit bigger than than x. 
Well, what that's really saying is that h 
is positive, right? 
I shift the input to the right a little 
bit. 
Well then, h*f'(x) is going to be 
positive because a positive number times 
a positive number is positive. 
And that means that f(x)+h*f'(x) will be 
bigger than f(x). 
We're just add something to both sides of 
this inequality. 
Now, f(x)+h*f'(x), that's about f(x)+h. 
So, although this argument isn't entirely 
precise yet, what it looks like it's 
saying is that the function's output at 
x+h is bigger than the function's output 
at x. 
So, if you plug in bigger inputs, you get 
bigger outputs. 
What about when the derivative is 
negative? 
We can play the same kind of game when 
the derivative's negative. 
Here we go. So again, x+h is just a bit 
bigger than x. 
And in that case, h is positive. 
But I've got a positive number times a 
negative number, h times the derivative 
of f is negative. 
Now, if I add f(x) to both sides, 
got that f(x)+h*f'(x) is less than f(x). 
But this is approximately the new output 
value of the function at x+h. 
So, I've got that the function's output 
at x+h is a little bit less than its 
output at f. 
So, a bigger input is giving rise to a 
smaller output. 
Even a little bit of information, whether 
or not the derivative is positive or 
negative, says something about the 
function. 
And you can see the same thing in your 
own life. 
For instance, suppose that the derivative 
of your happiness with respect to coffee 
is positive. 
What does that really mean? Well, that 
means that you should be drinking more 
coffee because an increase in coffee will 
lead to greater happiness. 
Of course, this is only true up to a 
point. 
After you've had a whole bunch of coffee, 
you might find that the derivative of 
your happiness, with respect to coffee, 
is zero. 
You should stop drinking coffee. 
Now, this makes sense because the 
derivative depends upon x, right? It 
depends upon how much coffee you've had. 
Not very much coffee, the derivative 
might be positive. But after a certain 
point, you might find that the 
derivative, vanishes. 
This seems like a silly example. 
coffee and happiness. 
But, so many things in our world are 
changing. 
And those changing things affect other 
things. 
The question is that when one of those 
things changes, does the other thing move 
in the same direction or do they move in 
opposite directions? And the sign, the S, 
I, G, N of the derivative records exactly 
that information. 
[MUSIC]