[SOUND] So as we proceed further with our understanding of derivative and
speed of change, sometimes we need to assume not only the speed
of change of a function, but the change of speed of derivative itself.
And you don't really realize how common this concept is in real life.
I am going to go with two basic examples.
First of all, it's example of inflation rates,
literally the speed of inflation, the speed of change of price's change.
And well, you'll always hear something that some politician
promises that rates would change slower.
He actually speaks about this speed of change of the derivative.
Another example is quite modern and quite important for us.
It's example of climate change.
And for example, let's take ocean level.
There is a common understanding that ocean level on average rises.
I'm not speaking about what is an average here and how it is calculated,
because it's kind of tricky and it's statistical answers.
Mark Twain said that there is three types of lies, lies, big lies, and
statistics, and well, we're not going to dive into that.
But what we are going to speak about, we're going to speak about
what's actually an understanding of global climate change here.
And the global climate change here, global warming here is presented as follows.
We do understand that ocean level rises and falls, well,
quite a lot throughout human history and all the plant history.
But then the question here is, how fast characteristically, and
what is the trend, it changes?
Well because if it changes, when you're for example,
growing linearly, it's not a big deal here, it's kind of common thing.
But if it is, for example, quadratic, faster than linear,
extremely faster than linear, then it is quite a big difference and
we need to do something with it.
Well spoiler alert, we need to do something with it.
And well, the mathematical definition of global warming
is the common class of a derivative of an ocean level.
Is the speed of change of ocean level linear or more, faster than linear?
So all these things ask, basically tackling the issue of this change of
the derivative of a derivative, the second derivative or the second order derivative.
First of all, lets us do some formal definition here.
We are considering our real-valued function f, and
we assume that it's differentiable, and we are having some given point a.
And so the second derivative is basically the derivative of
a derivative as is a limit where x approaches our given point a,
of fraction of change of derivative towards stage of argument.
Well, that's something which you are actually not glad to see,
because it's a definition, and we kind of hate all the complex definitions here.
But well, it works.
Well, in other words, as I said earlier, it's just a derivative
of the derivative, and it's nice and easy to calculate here.
So let us assume some example.
Well, first of all, let's start with a simple function like x squared and
come to some understanding of the second derivative here.
Now the first derivative here is, that's right, first derivative is 2x.
You all remember that, it's nice, and then we will need to find an algebra here.
And well, actually, I'm not going to ask you about it.
It's kind of obvious here, it's just 2, and everybody knows it.
So let us consider more complex examples here as exponents and sine function of x.
Well firstly, with exponent, that's quite easy,
because the derivative of exponent is exponent.
The second derivative is pretty much the same thing.
You can do anything with an exponent by taking a derivative.
And as for sine function, it's a bit tricky, but still impressively the same.
As is, the first derivative of sine function is cosine function,
thus the second derivative of sine function is the derivative
of cosine function, which is minus sine function.
So let us summarize for agreement here.
As for exponent, the second derivative of it is the same function, it holds still.
But as for the sine function, the second derivative is minus sine function,
minus the function itself.
It's kind of implying that sine function and exponents is connected somehow.
And if you study further complex analysis or
just basic complex numbers theorems, you will understand that
this connection is basically, or this rule for complex numbers.
And can be usually easily used by introducing imaginary part and
imaginary numbers into this equation.
So let us move further towards the motivation of all the work we're
doing here with second derivative.
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