[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]