[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]