Introduction to Derivatives or Idea of Derivatives Video Lecture | Mathematics (Maths) for JEE Main & Advanced

hi friends we are going to learn a new
sharpening calculus that's called
derivative up to this point we have
learned in detail the limit chopper
right once you have finished this all
limit now we are ready to learn the
derivative chopper right and again I'm
telling you before learning this
derivative I want to go back a little
bit back in Co the geometry chopper
right especially related to the slope so
suppose this is my y-axis and this is my
x-axis right this is x axis and this is
y axis and I have a line suppose this is
my line which is about say y equals to
MX plus C right MX plus C or I can also
write suppose it as a function write f
of X so now in this in this in this line
this am actually represents the slope
right slope as we have learned in the
coordinate geometry and we need to find
out the value of this slope and to find
the value of the slope of a line as we
are doing in the coordinate geometry we
need to have the two points right so I'm
taking this as a point and this has a B
point so let let me suppose the x
coordinate of this a point is X right so
the corresponding y coordinate will be f
of X right if you just plug the value of
x in this function FX then you'll get
the value of y coordinate right so the
corresponding y coordinate is f of X
that's actually this is a y coordinate
right and now here we are going to find
again the x coded up the B let me
suppose this B point is about H distance
R we write as the distance away from the
a point right so the total x coordinate
of this B point will be this X plus this
edge right this x + H therefore what
will get the corresponding y coordinate
here if you just plug this value x + H
and this function you'll get f of X plus
h that's the y coordinate of this p
point right so we get the x coordinate
so we get the coordinates of the a point
also and B points the coordinate of the
a point is simply the x coordinate is
actually
right X comma the white corn it is f of
X will be writing in the f of X from
right generally we used to do this in
the calculus now here the coordinate of
the b1 we get X plus s that's that's X
plus s that's the x-coordinate and the
y-coordinate we get f of X plus h f of X
plus h this is our y coordinate of the B
point right and in college symmetry to
find the slope of any line right f of
any line if you have the two points
given right two points call it given
then we used to use this formula slope
of this line will be M we will represent
this slope by M is simply the difference
of y coordinate that is y2 minus y1 over
difference of x coordinate that is x2
minus x1 or simply the change in Y to
the change in X right so now in this in
this and this point this will walk as a
y2 and this will work as an x2 here this
will work as a x1 and this will work as
a y1 so I'm just supposing this you can
choose any one right so y2 and y1 so y2
here is f of X plus h right - white y1
its y1 exactly right y1 y1 is the f of X
right over X 2 that is X plus h minus
minus x1 right so minus x1 x1 is here X
only simply X right
this is kawa ok don't suppose this is X
1 okay fine now here what we get we can
cut this X and X and finally we get the
slope as f of X plus h minus f of X over
H this is the slope of this line right
this y equals MX plus C but what about
in curve right we're going to find out
now the slope all right
slope of the curve line and before let
me tell you before this actually the
slope means just the rate of change you
know the rate of change of this function
this y equals MX plus C it's a function
right or the simply the rate of the
change of this line how this line is
changing how this function is changing
simply how the why party changing with
respect to the expert simply the change
of this function it it it it indicates
exactly right so same in the curve also
but ok fine for a particular line it's
constant right in a straight line
actually in a linear line it's constant
right this this this this part is
constant our slope is constant for us
linear line now we're going to learn
about the curved line right so let me
make the axis right suppose this is my x
axis and y axis I'm going to make a
curved line like this suppose this is my
curved line suppose this is function f
of X right f of X function again as we
will learn well previously to find to
find the slope of this or the rate of
change right simply how it is changing
we need to find out the two points I
mean we need to take the two points so
I'm taking this as suppose P point and
this has a Q point this is Q point right
and the x-coordinate of this P point let
me suppose X again right so the
corresponding y coordinate will be one I
plug the value of x here so I will get f
of X right and again here right
again I'm supposing in the same way that
it is about adds distance R we write
from the P point is so the x coordinate
of this Q point will be again X plus h
right so the corresponding y coordinate
of this Q point will be f of X plus h
right simply I will be plugging this X
plus s value in this function then I
will get the y coordinate right actually
this is why it means Y right so now
again what we got weaker the x
coordinate x coordinate as X right f of
X and the y f of X is the y coordinate
and here q also the corners of the cube
when we get that is X plus h X plus h
comma Y content that is f of X / set f
of X plus h right so to find the how are
you changing or to find the rate how it
is changing right
we need to take a secant line that is
passing through the P and Q this the
speaker this this PQ is a secant now the
secant now tells us the slope of this
secant now tells us how this has been
changed this function has been stained
what is the rate this right what is the
rate of change of this function between
P and Q point right so again I'm just
finding the slope of this function right
slope of this sorry secant so the slope
of secant slope of secant will be again
the same formula y2 minus y1 over x2
minus x1 so here y2 will be walking as f
of X plus h this will walk as y2 and y1
will walk as an f of X f of X whole
divided by x2 that is X plus h X plus h
minus X so again what I can do I can
simply cut this an X so I get the slope
M right
ok slope of the secant as f of X plus h
minus f of X whole divided by H right
since we cut this so we get the value of
the slope right now value of the slope
of this secant line right so this is the
rate again the rate again how this
function is changing between this P and
Q point so if I need to find out how we
change it between a P and suppose this Q
- point right Q - point so I will be
using this as x2 y2 here in this
function external y2 will be now the
coordinates of this Q - point call this
of this cube - points and at the time it
will be suppose H 1 and this H 1 will be
less than this edge right less than this
actual here it will be less than this
edge right
now if I want to find out here then it
will be queued as that suppose right so
again here the coded of this adds to
will be more less than H and as I will
come closer to this the value of H will
also decrease initially it is this much
bigger and more bigger more smaller more
smaller more smaller more smaller more
smaller more smaller means as I'm going
to closer to this P point the value of H
is gradually decreasing decreasing and
decreasing right let me show you here I
have a video about that see ya this is
as a P point right this will work as a P
point and this will work as a cue point
and what happened when I'm decreasing
the value of s means the distance of
this this x-coordinate and this
x-coordinate this is the this actually
this difference is the H actually right
the coordinate the difference of the x
coordinate here is the H it means this
and this right so what happens to this
this C right so this is q1 and people
now this I'm taking this right see ya
bringing it more closer more closer
means H is decreasing more decreasing
more decreasing more decreasing and I'm
going tozer and closer to this P point
more closer more closer more closer more
closer and I'm coming almost almost
close to this Q point and finally it is
going to be 0 right so let me do it once
again here that all right see I can find
I hope you understand this right yeah it
is another also video I think if you
wants this Right see ya this this here
the H is almost coming to 0 right almost
becoming 0 sorry it's almost the edge
part is almost becoming 0 and this this
Q point is almost going closer to this P
point but not exactly at the P point
means that H is not exactly equals to 0
but very very infinitely small to 0
right infinitely very close to 0 simply
almost close to 0 almost close to 0 but
not exactly the zero right so now we
need to find out the value of the slope
the secant slope we are going
to find out at very close to this P
point so to find out the the slope very
close to this P point we need to take
the limit right to get the precise level
right to get into the precise level we
to take the limit that we usually learn
so and how we have to take the limit
limit we have to take as s tends to 0 so
the hone ads tends to 0 this Q point
will almost comes almost Mars to this P
point and at that time right at that
time what it'll be when X tends to 0 the
slope of the secant write will be equals
to limit X tends to 0 f of X plus h
minus f of X all divided by H right this
this is the slope of the secant line at
that time and this is exactly the
derivative this is exactly called the
derivative means the slope of the secant
line at a particular point is exactly
the derivative right and again if you
wants to say in another way this still
beta again indicates the rate rate of
changing of this this rate of changing
right rate of changing this KUB between
between this that particular time that
that that particular point right how it
is changing it is changing an infinitely
small way right infinitely small really
changing but it is exactly the same
thing how this y part and the x part is
changing it's the ratio or you can say
the rate how the function is changing at
a particular point right at a particular
point in the curve how it is changing
right a particular point how the X Y
part right and X part ex parte changing
it's it's again the same meaning means
derivative simply will be meaning now
that derivative is just the slope of the
secant line at a particular point right
or the tangent at a particular point you
can say so it's going to be very longer
video right and I'm going to make more
much more video longer video because I'm
using the YouTube right so we'll be
meeting in the next video with lots of
concept again right
some more concept in the
right bye-bye