[MUSIC] Thus far, I've been trying to 
sell you on the idea that the derivative 
of f measures how we wiggling the input 
effects the output. 
A very important point is that 
sensitivity to the input depends on where 
you're wiggling the input. 
And here's an example. 
Think about the function f(x)=x^3. 
f(2) which is 2^3 is 8. 
f(2.01) 2.01 cubed is 8.120601. 
So, the input change of 0.01 was 
magnified by about 12 times in the 
output. 
Now, think about f(3) which is 3^3, which 
is 27. 
f(3.01) is 27.270901 so the input change 
of 0.01 was magnified by about 24 times 
as much, 
right? This input change and this input 
change were magnified by different 
amounts. 
You know, you shouldn't be too surprised 
by that right, the derivative, of course, 
measures this. 
The derivative of this function is 3x^2, 
so the derivative at two is 3*2^2 is 3*4 
is 12 and not coincidentally, there's a 
12 here and there's a 12 here, right, 
that's reflecting the sensitivity of the 
output to the input change. And the 
derivative of this function at 3 is 
3*3^2, which is 3*9 which is 27 and 
again, not too surprisingly here's a 27, 
right? The point is just that how much 
the output is effected depends on where 
you're wiggling the input. 
If you're wiggling around 2, the output 
is affected by about 12 times as much if 
we're wiggling around 3, the output is 
affected by 27 times as much, 
right? The derivative isn't constant 
everywhere, it depends on where you're 
plugging in. 
We can package together all of those 
ratios of output changes to input changes 
as a single function. 
What I mean by this, well, f'(x) is the 
limit as h goes to 0 of f(x+h)-f(x)/h. 
And this limit doesn't just calculate the 
derivative at a particular point. 
This is actually a rule, right, this is a 
rule for a function. 
The function is f'(x) and this tells me 
how to compute that function at some 
input X. 
The derivative is a function. 
Now, since the derivative is itself a 
function, I can take the derivative of 
the derivative. 
I'm often going to write the second 
derivative, the derivative of the 
derivative this way, 
f''(x). 
There's some other notations that you'll 
see in the wild as well. 
So, here's the derivative of f. 
If I take the derivative of the 
derivative, this would be the second 
derivative but I might write this a 
little bit differently. 
I could put these 2 d's together, so to 
speak, and these dx's together and then 
I'll be left with this. 
The second derivative of f(x). 
A subtle point here is if f were maybe y, 
you might see this written down and 
sometimes people write this dy^2, that's 
not right. 
I mean, it's d^2 dx^2 is the second 
derivative of y. 
The derivative measures the slope of the 
tangent line, geometrically. 
So, what does the second dreivative 
measure? Well, let's think back to what 
the derivative is measuring. 
The derivative is measuring how changes 
to the input affect the output. 
The deravitive of the derivative measures 
how changing the input changes, how 
changing the input changes the output, 
and I'm not just repeating myself here, 
it's really what the second derivative is 
measuring. 
It's measuring how the input affects how 
the input affects the output. 
If you say it like that, it doesn't make 
a whole lot of sense. 
Maybe a geometric example will help 
convey what the second derivative is 
measuring. 
Here's a function, y=1+x^2. 
And I've drawn this graph and I've 
slected three points on the graph. 
Let's at a tangent line through those 3 
points. 
So, here's the tangent line through this 
bottom point, the point 0,1 and the 
tangent line to the graph at that point 
is horizontal, right, the derivative is 0 
there. 
If I move over here, the tangent line has 
positive slope and if I move over to this 
third point and draw the tangent line 
now, the derivative there is even larger. 
The line has more slope than the line 
through that point. 
What's going on here is that the 
derivative is different. 
Here it's 0, here it's positive, here 
it's larger still, right? The derivative 
is changing and the second derivative is 
measuring how quickly the derivative is 
changing. 
Contrast that with say, this example of 
just a perfectly straight line. 
Here, I've drawn 3 points on this line. 
If I draw the tangent line to this line, 
it's just itself. 
I mean, the tangent line to this line is 
just the line I started with, right? So, 
the slope of this tangent line isn't 
changing at all. 
And the second derivative of this 
function, y=x+1, really is 0, right? 
The function's derivative isn't changing 
at all. 
Here, in this example, the function's 
derivative really is changing and I can 
see that if I take the second derivative 
of this, 
if I differentiate this, I get 2x, and if 
I differentiate that again, I just get 
two, which isn't 0. 
There's also a physical interpretation of 
the second derivative. 
So, let's call p(t), the function that 
records your position at time t. 
Now, what happens if I differentiate 
this? What's the derivative with respect 
to time of p(t)? 
I might write that, p'(t). 
That's asking, how quickly is your 
position changing, well, that's velocity. 
That's how quickly you're moving. 
You got a word for that. 
Now, I could ask the same question again. 
What happens if I differentiate velocity, 
I am asking how quickly is your velocity 
changing. 
We've got a word for that, too. 
That's acceleration. 
That's the rate of change of your rate of 
change. 
There's also an economic interpretation 
of the second derivative. 
So, maybe right now dhappiness, ddonuts 
for me is equal to 0, 
right? What this is saying? This is 
saying how much will my happiness be 
affected, if I change my donut eating 
habits. 
If I were really an economist I'd be 
talking about marginal utility of donuts 
or something, but, this is really a 
reasonable statement, right? This is 
saying that right at this moment you 
know, eating more donuts really won't 
make me any more happier and I probably 
am in this state right now, because if 
this weren't the case, I'd be eating 
donuts. 
So, let's suppose this is true right now 
and now, something else might be true 
right now. 
I might know something about the second 
derivative of my happiness with respect 
to donuts. 
What is this saying? Maybe this is 
positive right now. 
This is saying that a small change to my 
donut eating habits might affect how, 
changing my donut habits would affect how 
happy I am. 
If this were positive right now, should I 
be eating more donuts, even though 
dhappiness, ddonuts is equal to zero? 
Well, yeah, if this is positive, then a 
small change in my donut eating habits, 
just one more bite of delicious donut 
would suddenly result in dhappiness, 
ddonuts being positive, 
which should be great, then I should just 
keep on eating more donuts. 
Contrast this with the situation of the 
opposite situation, where the second 
derivative happens with respect to donuts 
isn't positive, but the second derivative 
of happiness with respect to donuts is 
negative. 
If this is the case I absolutely should 
not be eating any more donuts because if 
I start eating more donuts, then I'm 
going to find that, that eating any more 
donuts will make me less happy. 
Let's think about this case 
geometrically. 
So here, I've drawn a graph of my 
happiness depending on how many donuts 
I'm eating. 
And here's two places that I might be 
standing right now on the graph. 
These are two places where the derivative 
is equal to zero. 
And I sort of know that I must be 
standing at a place where the derivative 
is 0, because if I were standing in the 
middle, I'd be eating more donuts right 
now. 
So, I know that I'm standing either right 
here, say, or right here. 
Or maybe here, or here. 
I'm standing some place where the 
derivative vanishes. 
Now, the question is how can I 
distinguish between these two different 
situations? Right here, if I started 
eating some more donuts, I'd really be 
much happier. 
But here, if I started eating some more 
donuts I'd be sadder. 
Well, look at this situation, this is a 
situation where the second derivative of 
happiness to respected donuts is 
positive, 
right? 
When I'm standing at the bottom of this 
hole, a small change in my donut 
consumption starts to increase the extent 
to which a change in my donut consumption 
will make me happier, 
alright? If I find that the second 
derivative of my happiness with respect 
to donuts is positive, I should be eating 
more donuts to walk up this hill to a 
place where I'm happier. 
Contrast that with a situation where I'm 
up here. 
Again, the derivative is zero so a small 
change in my doughnut consumption doesn't 
really seem to affect my happiness. 
But the second derivative in that 
situation is negative. 
And what does that mean? That means a 
small change to my donuts consumption 
starts to decrease the extent to which 
donuts make me happier. 
So, if I'm standing up here and I find 
that the second derivative of my 
happiness with respect to donuts is 
negative, I absolutely shouldn't be 
eating anymore donuts. 
I should just realize that I'm standing 
in a place where, at least for small 
changes to my donut consumption, I'm as 
happy as I can possibly be and I should 
just be content to stay there. 
There's more to this graph. 
Look at this graph again. 
So, maybe I am standing here. 
Maybe the derivative of my happiness with 
respect to donuts is zero. 
Maybe the second derivative of my 
happiness with respect to donuts is 
negative. 
So, I realize that I'm as happy as I 
really could be for small changes in my 
donut consumption. 
But if I'm willing to make a drastic 
change to my life, 
if I'm willing to just gorge myself on 
donuts, things are going to get real bad, 
but then they're going to get really 
really good and I'm going to start 
climbing up this great hill. 
It's not just about donuts, it's also 
true for Calculus. 
Look, right now, you might think things 
are really good, they're going to get 
worse. But with just a little bit more 
work, you're eventually going to climb up 
this hill and you're going to find the 
immeasurable rewards that increased 
Calculus knowledge will bring you. 
[MUSIC]