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[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).

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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.

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