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[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.

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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.

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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]