We are going to talk about Derivatives today.
Derivative itself is a limit. It is a limit
of ratios. It measures the change of x, change
of the function of the f related to the change of the independent variable x. So, f dash of x
is a limit as h tends to 0, this is for
a general function for what to or so, I take
a function from r to r, whose real line to
the real line and I am giving the definition
of the derivative. Now, you see this is the
change in the function, well this is the change
in the value of the independent variable x
has change to x.
Now, as I told you that h when come to 0 from
both this sides, this side or this side. So,
you come to these from right side you say,
is which is going to 0 plus. If you come from
left side you will say which is coming from
0 minus.
Now, what is further important to note is
that this limit has to exist and if this limit
exist it means it has two components, the
right derive right limit and the left limit
and both should exist. The right limit of
this is called the right derivative; the left
limit of this is called the left derivative,
which we will write as follows, the right derivative.
So, right derivative is a h tends
to 0 plus and the left derivative.
Now, derivative exists
if these two limits exist; left and right
limit and they must be equal. The derivative
exist only under this condition the left derivative
equals the right derivative and which the
under common value is the value of the derivative.
Of course you know of several facts, now what about a function f from open into our a b
to r and what about a function f from closed interval a b to r. You observe that when you
are taking open interval, the points a and
b are not in the interval and hence
if you take any point in a and b here or here, however, near they are towards a or b we can still
have a both sided limit that are essentially
your call the in the quantum of the real line.
In the sense of infinite of the real line
these certain things the real line has to,
we felt it cannot really be thought and now
there are some remark that I want to make
when you are talking about this what, when
I say the derivative exists I say on for a
function from a close interval a 2 b 2 means
f dash x exist finitely for all x in the open
interval part. Now, when I am talking, what
do when I am talking about a derivative at
a I am actually then considering the right
derivative at a and the left derivative at
b. This is very, very fundamental issue, but
one has to be very clear in his mind from
the very beginning.
Now, you are all experienced about derivatives,
you have learned it in calculus. You all know
that this simply means where n is a real number.
So, if I take the derivative of this you know
how to calculate it through first principles,
but I want to tell you that the sudden must
interesting use of the derivative which will
soon come, but let me tell you something,
for example, if you want to compute a derivative
of x is equal to mode x does it have a derivative
everywhere absolute value x, answer is no;
so will show that at x equal to 0, f x is
equal to mode of x has no derivative.
So, when I am talking about a derivative of
a function over the whole domain or the whole
or makes meaning that each at every each and
every point the derivative exist. So, these
things has to be understood when these are very simple things. Now, how do I know that
the derivative we have taken at x equal to
0 does not exist? This is by checking the
right derivative and the left derivative.
So, for this function the right derivative
is given in this way. At x equal to 0, my
x is now 0. So, I am replacing x with 0, 0
plus h minus 0 by h, now this mode of h y
h, because h is positive. So, h mode of h
is h. So, h by h is one left derivative let
us calculate.
Now, because if since h is now will strictly
less than 0 mode of h is minus h. So, minus
1 is if those who have forgotten what is mode
of x, let me tell you mode of x is equal to
x if x is greater than or equal to 0 is equal
to minus x, if x is strictly less than 0 then
if x is a negative one, if you only get its
positive plot that is meaning of modular x
that is the mode x simply means the distance
of a number from 0. If you have 0 you to have
2 and minus 2 the distance of if you are measuring
centimeters each block 1 centimeter then distance
of 0 to 2 is 2 centimeter and distance from
0 to minus 2 is 2 centimeter.
Every function need not have a derivative
and if you look at f x is equal to mode x
you look at its diagram f x is absolute value
of x then you see these continuous function
because you can draw the graph of this function
without lifting your pen from the paper. So,
again I want to go back to Renova Mendeleev's
about statement that you would know calculus
much better, if you know at the very outside
at every continuous function did not have
derivative of course, they have functions
which are continuous at every point and not
differentiable at any point this a monster
functions by Weierstrass which we are not
going to make any discussion.
There is an interesting idea called the symmetric
derivative this is just for I am talking about
I am really not going to deal with it much.
It is called the symmetric derivative. In
symmetric derivative at x is given as follows
limit h tends to 0 f of x plus h minus f of
x minus h. So, I am looking at a two n points
both movement this side and that side divided
by twice of h. Now, the funny part is that
these called a symmetric derivative, now the
funny part is that this derivative can exists
even the derivative itself do not exist, for
example, let us now consider f of x is equal
to mode of x and let us try to calculate its
symmetric derivative. It is limit h tends
to 0 that is 0 plus mode h minus 0 minus mode
h by h. So, you can have two parts, right.
So, limit h tends to 0 even divided into two
parts, the h positive h negative and left
limit, left limit, left limit, right limit
and do it, but that is not really require
thus this is mode h minus mode of minus h.
So, mode of minus h is same as mode h, basically
this is 0 because this same as limit mode
h minus mode h. If h is positive this is negative,
this will again have minus and become mode
h. So, we can do the same thing if h is positive
say this is this is negative, there will be
minus in front of it and it will become h.
It will become h minus h you can try it out
with h negative separately if you are not
convince with the fact that mode minus h is
minus h because the distance of h from 0 and
distance of minus h from 0. We got the same
we just have to understand that the absolute
value of a given number is nothing, but it
distance from the 0 and distance from the
origin 0. So, this is 0.
The symmetric derivative of this function
exist even though symmetric derivative is
not when the actual derivative does not exist.
In fact, if f has a derivative then the symmetric
derivative is equal to the derivative that
is which are the points. Let us now look into
the uses.
For example, yesterday we were talking about
a limit of this form and will see how useful
it becomes, now let us try to look at the
function f of x is equal to log of x of course,
x has to be greater than 0 that is a domain.
The domain of this function, basically this
function f log x this log function it actually
defines from 0. So, you know already that
the domain of this function is 0 to infinite.
Now, we want compute this from the first principles
at when want to compute this limit. So, this
is same as limit h tends to 0. So, I write
again.
So, what I will get is h log of 1 plus h by
x see my x is now fixed 
x is a fixed number and x cannot be 0 x has
to be bigger than 0. Now, what I will do is
here my I have one limit h by x. So, I will
divide, multiply both sides top and bottom
by one by x. So, I will have limit h tends
to 0 one by x multiplied at the top which
can be brought out where it just a constant
it is not depending on it h by x.
Now, observe that on when h is going to 0
and x is a fixed positive number h by x also
goes to 0. So, this is same as 1 by x limit
h by x going to 0 log of 1 plus h by x by
h by x and that is known to be 1. So, ultimately
we have 1 by x, this is a very, very useful
way to obtain the limit you see how we are
made a use of this limit, but now you will
see we have never given a proof to how do
I actually get this limit though we have used
it.
Here, interesting way to prove it is, to use
what is called the L' Hospital's Rule. In
India we will call it L’ hospital rule,
but if you go by the French pronunciation
it is called L' Hospital's Rule, he was taught
by Jon Bernoulli and possibly the result was
given by Jon Bernoulli to L' Hospital's Rule
was a Richman Sonin, he later on wrote a book
on the calculus.
Now, what is a L' Hospital's Rule? L' Hospital's
Rules essentially deals with trying to find
limits of this form now suppose f and g are
both nice continuous functions you do functions
got, but suppose f of a by g of a is in the
from 0 by 0 then of course, you cannot say
I will just take limit f x by limit g x tends
not possible because its 0 by 0. So, your
divisibility rule that you have about limits
completely breaks down here right. What should
be done now handle such limits. So, that this
interesting rule its say that it is if f dash
x and g dash x exists and if when if the functions
are differentiable.
Of course, we have to assume needs differentiable
at every x and if g dash x is not equal to
0 then limit of x tends to a f x by g x is
equal to limit of f dash x by g dash x and
when I am talking about f dot x g dot x exist
means I am talking about that the derivative
exist at all x. So, from the domain of the
function and then if g dash x is not equal
to 0 on that domain for every x then this
limit is same as this limit what could be
a possible application of this let us see.
Let us look at the 
following limit again, if I put x equal to
0 that is sin 0 y 0 if this is again in the
0 y 0 form, but I know the derivative of sin
exist sin of x if I take the derivative which
we write at as like this the derivative is
also a just for your recalling because you
know it in high school the derivative of if
I take also written as f dash x, this was
a notation of Newton.
While Olier give the following notation some
people also credited, but Olier is really
the person who gave this made this notation
popular where we call it d d x of the function
f never usually we say d f d x or d f by d
x. It is not a way d is not a ratio d f is
not something and d s is not something its
d d x is an operator which is operating on
f.
So, you have to remember you should always
call d d x of f that is the right thing which
we are basically we adding just the short
end to right if very fast is not d f d x is
is not a ratio of two numbers. So, d d x of
sin x is cosine of x right and when cosine
of x is not equal cosine of x is there. It
exists and d d x of x is 1 and that is not
equal to 0, which means limit of x tends to
0 sin x by x is same as limit of x tends to
0 cos x by 1 and as x tends to 0 cos 0 is
continuous function and that is what you get.
Then let us look as a last example for today
is the logarithmic is the one which we are
just used to compute the derivative of log
x. Now, if I put x equal to 0 it will become
log 1. So, log is 0, it will again in a 0
by 0 form. So, let me did the derivative of
log x now I am a see some sort of a little
bit of circular thing of course, there is
a geometrical proof this you can do a proof
this, but we are not going to do a proof of
this of course, these limits came into existence
when we were this people when they have first
got to this limits they actually did it by
computation did it by experimentation.
Remember, mathematics is also an experimental subject, but thing which I tried to tell you
in the last lecture and very specifically
so is calculus. Now, if you take the derivative
of this, it is 1 by 1 plus x and you know
if x is strictly greater than 0 these also
known 0 this is the very well defined thing
and derivative of x. So, d d x of log x is
this and d d x of x x is 1. Basically, now
again by applying the L' Hospital's Rule or
L’ Hospital, if you want then we have limit
x tends to 0 log of 1 plus x by x as 1 by
one plus x limit x tends to 0 by 1. Now, these
are quantum function where x is strictly greater
than 0 an x is tends into 0 and that gives
you 1.
With this we end today’s talk and tomorrow
will tell you some more important properties
of the derivative, how to differentiate sum
of 2 functions, the product 2 functions, the
cosine 2 functions, the ratio of 2 functions
and some of their more interesting properties.
So, with this we end out of today.
Thank you very much.
