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[MUSIC] In the future, we're going to 
have a lot of very precise statements 

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about the derivative. But before we get 
there, I want us to have some intuition 

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as to what's going on. 
Let's take a look at just the S, I, G, N, 

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the sign of the derivative. 
The thick green line of the plot is some 

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random function, and the thin red line is 
its derivative. And note that when its 

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derivative is positive, the function's 
increasing, and when the derivative is 

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negative, the function's decreasing. 
We can try to explain what we're seeing 

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here formally, where that calculation on 
paper. 

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So let's suppose that the derivative is 
positive over a whole range of values. 

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And, we also know something about how the 
derivative is related to the functions 

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values. 
The function's output or x+h is close to 

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the functions output of x plus how much 
the derivative tells us the output should 

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change by, which is how the input changed 
by times the ratio change of output 

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change to input change. 
Alright. Now let's suppose that x+h is a 

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bit bigger than than x. 
Well, what that's really saying is that h 

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is positive, right? 
I shift the input to the right a little 

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bit. 
Well then, h*f'(x) is going to be 

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positive because a positive number times 
a positive number is positive. 

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And that means that f(x)+h*f'(x) will be 
bigger than f(x). 

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We're just add something to both sides of 
this inequality. 

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Now, f(x)+h*f'(x), that's about f(x)+h. 
So, although this argument isn't entirely 

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precise yet, what it looks like it's 
saying is that the function's output at 

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x+h is bigger than the function's output 
at x. 

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So, if you plug in bigger inputs, you get 
bigger outputs. 

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What about when the derivative is 
negative? 

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We can play the same kind of game when 
the derivative's negative. 

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Here we go. So again, x+h is just a bit 
bigger than x. 

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And in that case, h is positive. 
But I've got a positive number times a 

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negative number, h times the derivative 
of f is negative. 

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Now, if I add f(x) to both sides, 
got that f(x)+h*f'(x) is less than f(x). 

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But this is approximately the new output 
value of the function at x+h. 

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So, I've got that the function's output 
at x+h is a little bit less than its 

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output at f. 
So, a bigger input is giving rise to a 

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smaller output. 
Even a little bit of information, whether 

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or not the derivative is positive or 
negative, says something about the 

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function. 
And you can see the same thing in your 

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own life. 
For instance, suppose that the derivative 

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of your happiness with respect to coffee 
is positive. 

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What does that really mean? Well, that 
means that you should be drinking more 

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coffee because an increase in coffee will 
lead to greater happiness. 

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Of course, this is only true up to a 
point. 

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After you've had a whole bunch of coffee, 
you might find that the derivative of 

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your happiness, with respect to coffee, 
is zero. 

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You should stop drinking coffee. 
Now, this makes sense because the 

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derivative depends upon x, right? It 
depends upon how much coffee you've had. 

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Not very much coffee, the derivative 
might be positive. But after a certain 

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point, you might find that the 
derivative, vanishes. 

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This seems like a silly example. 
coffee and happiness. 

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But, so many things in our world are 
changing. 

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And those changing things affect other 
things. 

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The question is that when one of those 
things changes, does the other thing move 

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in the same direction or do they move in 
opposite directions? And the sign, the S, 

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I, G, N of the derivative records exactly 
that information. 

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[MUSIC]