[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. [SOUND]