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[MUSIC] Sometime in the future, we're 
going to see the following derivative 

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rule. 
But, I want to mention it now just so we 

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can see an example of how derivatives 
play out in practice. 

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The derivative of the √x is 1/2√x. 
You might already believe this if you 

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believe the power rule, 
right? 

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The derivative of x^n is nx^n-1. So, if n 
is 1/2, then I've got that the derivative 

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of x^n, now 1/2, is n, 1/2, times x^n-1. 
And conveniently, 1/2-1=-1/2. 

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This is really the same as this, it's 
just written here with the square root 

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symbol instead of with the exponents. 
We can use this derivative rule to help 

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explain certain numerological 
coincidences. 

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Let's take a look. 
Look, the √9999 is 99.9949998, okay, so 

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it keeps on going forever. 
It's irrational. 

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But, this is bizarrely close to 99.99 
instead of 499 just saying it's close to 

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99.995. 
Is this just a coincidence? This isn't a 

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coincidence. 
Look. 

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The square √10000 is 100 because 100^2 is 
10,000. 

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What I'm really doing here is wiggling 
the input. 

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I'm going from 10,000 to 9,999. 
In other words, I'm trying to calculate 

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the √10000-1, wiggling the input down a 
bit. 

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What does the derivative calculate? Well, 
the derivative calculates the ratio 

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between output change to input change. 
So, the √10000 wiggled down a little bit 

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is about the √10000 minus how much I 
change the input by times the ratio of 

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how much I expect the output change 
compared to the input change. 

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Now, we can try to calculate the 
derivative at 10,000. 

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What's the derivative at 10,000? Well, 
it's 1/2√10000. 

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The √10000 is 100, so it's 1/2*100. 
1/2*100 is 1/200, which is 0.005. 

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Look, the √9999 is so close to 99.995 
because the √10000 is 100. 

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And, when I shift the input down by one, 
this derivative calculation is suggesting 

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that the output should be shifted down by 
about 0.005, and indeed it is. 

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This is a great example of calculus. 
Yes, you could have asked your calculator 

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to compute the √9999, but you couldn't 
have asked your calculator to tell you 

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why. Why is that answer so mysteriously 
close to 99.995? 

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In short, calculus is more than 
calculating. 

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It's not about answers, it's about 
reasons. It's about explanations about 

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the stories that human beings can tell to 
each other about why that number and not 

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another. 
But, that's not say that the numbers 

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aren't fun to play with themselves, and 
we can use this same trick to do other 

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amazing feats like · , we can try to 
estimate the √82. 

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I know the √81 is 9. 
I'm trying to say something about the 

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√82. 
I'm trying to wiggle the input up a 

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little bit. 
Well, derivatives have something to say 

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about that. 
The √81+1, the √82, would be about the 

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√81, which is 9, plus how much I expect 
the output to change. 

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I wiggled the input, 
I expect the output to change by some 

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amount. 
Well, the derivative is measuring how 

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much I expect the output to change by. 
So, I'm going to take the derivative of 

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the function at 81, at the square root 
function at 81, I'm going to multiply by 

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how much I'm wiggling the input by. This 
will be how much I expect the output to 

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change when I change the input. 
Now, in this specific case, what's the 

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derivative at 81? Well, that's 1/2√81, 
which is 1/2*9. The √81 is 9, which is 

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1/18. So, I would expect the √82 to be 
about 9+1/18 because I expect wiggling 

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the input up to wiggle the output up by 
about 1/18. 

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And this is pretty good. 
There's actually two different ways to 

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tell if this isn't such a bad guess. 
Here's here's one way to tell. 

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What's what's 1/18? Well, it's 0.05 
repeating. 

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And, what's the actual value of the √82? 
It's 9.055. 

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Look, it's pretty close to 9 plus this. 
That's pretty good. 

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Another way to see that this isn't such a 
bad guess is just to take 9+1/18, and 

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square it. 
When I square 9+1/18, I get 

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9^2+2*9/18+1/1/8^2. 
2*1/2=1. 

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This is 81+1=82, and 1/18^2 is the very 
small number 1/324. 

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So, either way you look at it, we're 
doing pretty good to guess the √82 is 

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about this, 
and we're doing it with derivatives. 

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Again, it's derivatives for the win. 
By relating the input change to the 

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output change, we're able to estimate the 
values of functions that would be very 

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hard to access directly. 
[MUSIC]