[MUSIC] Sometime in the future, we're 
going to see the following derivative 
rule. 
But, I want to mention it now just so we 
can see an example of how derivatives 
play out in practice. 
The derivative of the √x is 1/2√x. 
You might already believe this if you 
believe the power rule, 
right? 
The derivative of x^n is nx^n-1. So, if n 
is 1/2, then I've got that the derivative 
of x^n, now 1/2, is n, 1/2, times x^n-1. 
And conveniently, 1/2-1=-1/2. 
This is really the same as this, it's 
just written here with the square root 
symbol instead of with the exponents. 
We can use this derivative rule to help 
explain certain numerological 
coincidences. 
Let's take a look. 
Look, the √9999 is 99.9949998, okay, so 
it keeps on going forever. 
It's irrational. 
But, this is bizarrely close to 99.99 
instead of 499 just saying it's close to 
99.995. 
Is this just a coincidence? This isn't a 
coincidence. 
Look. 
The square √10000 is 100 because 100^2 is 
10,000. 
What I'm really doing here is wiggling 
the input. 
I'm going from 10,000 to 9,999. 
In other words, I'm trying to calculate 
the √10000-1, wiggling the input down a 
bit. 
What does the derivative calculate? Well, 
the derivative calculates the ratio 
between output change to input change. 
So, the √10000 wiggled down a little bit 
is about the √10000 minus how much I 
change the input by times the ratio of 
how much I expect the output change 
compared to the input change. 
Now, we can try to calculate the 
derivative at 10,000. 
What's the derivative at 10,000? Well, 
it's 1/2√10000. 
The √10000 is 100, so it's 1/2*100. 
1/2*100 is 1/200, which is 0.005. 
Look, the √9999 is so close to 99.995 
because the √10000 is 100. 
And, when I shift the input down by one, 
this derivative calculation is suggesting 
that the output should be shifted down by 
about 0.005, and indeed it is. 
This is a great example of calculus. 
Yes, you could have asked your calculator 
to compute the √9999, but you couldn't 
have asked your calculator to tell you 
why. Why is that answer so mysteriously 
close to 99.995? 
In short, calculus is more than 
calculating. 
It's not about answers, it's about 
reasons. It's about explanations about 
the stories that human beings can tell to 
each other about why that number and not 
another. 
But, that's not say that the numbers 
aren't fun to play with themselves, and 
we can use this same trick to do other 
amazing feats like · , we can try to 
estimate the √82. 
I know the √81 is 9. 
I'm trying to say something about the 
√82. 
I'm trying to wiggle the input up a 
little bit. 
Well, derivatives have something to say 
about that. 
The √81+1, the √82, would be about the 
√81, which is 9, plus how much I expect 
the output to change. 
I wiggled the input, 
I expect the output to change by some 
amount. 
Well, the derivative is measuring how 
much I expect the output to change by. 
So, I'm going to take the derivative of 
the function at 81, at the square root 
function at 81, I'm going to multiply by 
how much I'm wiggling the input by. This 
will be how much I expect the output to 
change when I change the input. 
Now, in this specific case, what's the 
derivative at 81? Well, that's 1/2√81, 
which is 1/2*9. The √81 is 9, which is 
1/18. So, I would expect the √82 to be 
about 9+1/18 because I expect wiggling 
the input up to wiggle the output up by 
about 1/18. 
And this is pretty good. 
There's actually two different ways to 
tell if this isn't such a bad guess. 
Here's here's one way to tell. 
What's what's 1/18? Well, it's 0.05 
repeating. 
And, what's the actual value of the √82? 
It's 9.055. 
Look, it's pretty close to 9 plus this. 
That's pretty good. 
Another way to see that this isn't such a 
bad guess is just to take 9+1/18, and 
square it. 
When I square 9+1/18, I get 
9^2+2*9/18+1/1/8^2. 
2*1/2=1. 
This is 81+1=82, and 1/18^2 is the very 
small number 1/324. 
So, either way you look at it, we're 
doing pretty good to guess the √82 is 
about this, 
and we're doing it with derivatives. 
Again, it's derivatives for the win. 
By relating the input change to the 
output change, we're able to estimate the 
values of functions that would be very 
hard to access directly. 
[MUSIC]