JEE Exam > JEE Videos > Mathematics (Maths) Class 12 > Differentiability of a Function
Differentiability of a Function Video Lecture | Mathematics (Maths) Class 12 - JEE
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FAQs on Differentiability of a Function Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What does it mean for a function to be differentiable? |  |
Ans. A function is said to be differentiable at a point if the derivative of the function exists at that point. In simpler terms, it means that the function has a well-defined instantaneous rate of change at that specific point.
| 2. How can I determine if a function is differentiable at a given point? | |
Ans. To determine if a function is differentiable at a given point, you need to check if the left-hand derivative and the right-hand derivative exist and are equal at that point. If they are equal, then the function is differentiable at that point. However, if they are not equal, or if one or both derivatives do not exist, then the function is not differentiable at that point.
| 3. Can a function be differentiable everywhere except at a single point? | |
Ans. No, if a function is not differentiable at a single point, it is not differentiable in the entire domain. Differentiability requires the existence of derivatives at each point in the function's domain. If there is a single point where the derivative does not exist, then the function is not differentiable anywhere.
| 4. Is every continuous function differentiable? | |
Ans. No, not every continuous function is differentiable. Although all differentiable functions are continuous, the converse is not true. There are continuous functions that are not differentiable at certain points, such as functions with sharp corners or discontinuities in their graphs.
| 5. Can a function be differentiable but not continuous? | |
Ans. No, a function cannot be differentiable at a point if it is not continuous at that point. Differentiability implies continuity. If a function is not continuous at a certain point, it cannot have a derivative at that point, and therefore, it is not differentiable at that point.
Text Transcript from Video
what we're going to do in this video is
explore the notion of differentiability
at a point and that is just a fancy way
of saying does the function have a
defined derivative at a point so let's
just remind ourselves a definition of a
derivative and there's multiple ways of
writing this for the sake of this video
I'll write it as the derivative of our
function at Point C this is Lagrangian
with this F prime the derivative of our
function f at C is going to be equal to
the limit as X approaches C of f of X
minus f of C over X minus C and at first
when you see this formula we've seen it
before it looks a little bit strange but
all it is is it's calculating the slope
this is our change in the value of our
function or you could think of it as our
change in Y if Y is equal to f of X and
this is our change in X and we're just
trying to see well what what is that
slope as X gets closer and closer to C
as our change in X gets closer and
closer to zero and we talk about that in
other videos so I'm now going to make a
few claims in this video and I'm not
going to prove them rigorously there's
another video that will go a little bit
more into the proof direction but this
is more to get an intuition and so the
first claim that I'm going to make is if
f is differentiable at x equals C at x
equals C then F is continuous at x
equals C so I'm saying if we know it's
differentiable if we can find this limit
if we can find this derivative at x
equals C then our function is also
continuous at x equals C it doesn't
necessarily mean the other way around
and actually we'll look at a case where
it's not necessarily the case the other
way around that if you're continuous
then you're definitely differentiable
but another way to interpret what I just
wrote down is if you are not continuous
then you definitely will not be
differentiable if s not continuous
at x equals C then f is not
differentiable differentiable at X is
equal to C so let me give a few examples
of a non continuous function and then
think about would we be able to find
this limit so the first is where you
have a discontinuity our function is
defined at C it's equal to this value
but you can see as X becomes larger than
C it just jumps down and shifts right
over here so what would happen if you
are trying to find this limit well
remember all this is is the slope of a
line between when X is some arbitrary
value let's say it's out here so that
would be X this would be the point X
comma f of X and then this is the point
C comma F of C right over here so this
is C comma F of C so if you find the
left-sided limit right over here you're
essentially saying okay let's find this
slope and then let me get a little bit
closer and let me put let's get X a
little bit closer and then let's find
the slope and then let's get X even
closer than that and find this slope and
in all of those cases it would be 0 the
slope is 0 so one way to think about it
the derivative or this limit as we
approach from the left seems to be
approaching zero but what about if we
were take X's to the right so instead of
our exes being there what if we were to
take x's right over here well for this
point X comma f of X our slope if we
take f of X minus f of C over X minus C
that would be the slope of this line if
we get X to be even closer let's say
right over here then this would be the
slope of this line if we get even closer
then this expression would be the slope
of this line and so as we get closer and
closer to X being equal to C
we see that our slope is actually
approaching negative infinity and most
importantly it's approaching a very
different value from the right this
expression is approaching a very
different value from the right as it is
from the left and so in this case this
this limit up here won't exist so we can
clearly say this is not differentiable
so once again not a proof here I'm just
getting an intuition for if something
isn't continuous it's pretty clear at
least in this case that it's not going
to be differentiable let's look at
another case let's look at a case where
we have what sometimes called a
removable discontinuity or a point
discontinuity so once again let's say
we're approaching from the left this is
X this is the point X comma f of X now
what's interesting is where as this
expression is the slope of the line
connecting X comma f of X and C comma f
of C which is this point not that point
remember we have this removable
discontinuity right over here and so
this would be this expression is
calculating the slope of that line and
then if X gets even closer to C well
then we're going to be calculating the
slope of that line if X gets even closer
to C we're going to be calculating the
slope of that line and so as we approach
from the left as X approaches C from the
left we actually have a situation where
this expression right over here is going
to approach negative infinity and if we
approach from the right if we approach
with X is larger than C well this is our
X comma f of X so we have a positive
slope and then as we get closer it gets
more positive more positive approaches
positive infinity but either way it's
not approaching a finite value and one
side is approaching positive infinity or
the other side's approaching negative
infinity this the limit of this
expression is not going to exist so once
again I'm not doing a rigorous proof
here but try to construct a
discontinuous function where you will be
able to find this it is very very hard
and you might say well what about the
situations where s does not even defined
at C which for sure you're not going to
be continuous if f is not defined at C
well if F is not defined at C then this
part of the expression
wouldn't even make sense so you
definitely wouldn't be differentiable
but now let's ask another thing I've
just given you good arguments for when
you're not continuous you're not going
to be differentiable but can we make us
another claim that if you are continuous
then you definitely will be
differentiable well it turns out that
there are for sure many functions an
infinite number of functions that can be
continuous at sea but not differentiable
so for example this could be an absolute
value function it doesn't have to be an
absolute value function but this could
be Y is equal to the absolute value of X
minus C and why is this one not
differentiable at C well think about
what's happening think about this
expression remember this expression all
it's doing is calculating the slope
between the point X comma f of X and the
point C comma f of C so if X is say out
here this is X comma f of X it's going
to be calculate and as we're so as we
take the limit for as X approaches C
from the left we'll be looking at this
slope and then as we get closer we'll be
looking at this slope which is actually
going to be the same in this case it
would be a negative 1 so as X approaches
C from the left this expression would be
negative 1 but as we as X approaches C
from the right this expression is going
to be 1 the slope of the line that
connects these points is 1 the slope on
the line that connects these points is 1
so our the limit of this expression or I
would say the value of this expression
is approaching two different values as X
approaches C from the left to the right
from the left it's approaching negative
1 or it has constantly negative 1 and
until it's approaching negative 1 you
could say and from the right it's 1 and
it's approaching 1 the entire time and
so we know if you're approaching two
different values from on the left-sided
or the right-sided limit then this limit
will not exist so here this is not not
differentiable and even intuitively we
think of the derivative as the slope of
the tangent line and you could actually
draw an infinite number of tangent lines
here's one way to think about it you
could say well maybe this is the tangent
line right over there but why can't I
make something like
this the tangent line that only
intersects at the point C comma zero and
then you could keep doing things like
that why can't that be the tangent line
and you could go on and on and on so the
big takeaways here at least intuitively
and then in a future video I'm going to
prove to you that if F is differentiable
at C then it is continuous at C which
can also be interpreted in that if
you're not continuous at C then you're
not going to be differentiable these two
examples will hopefully give you some
intuition for that but it's not the case
that if something is continuous that it
has to be differentiable it oftentimes
will be differentiable but it doesn't
have to be differentiable and this
absolute value function is an example of
a continuous function at C but it is not
differentiable at C
explore the notion of differentiability
at a point and that is just a fancy way
of saying does the function have a
defined derivative at a point so let's
just remind ourselves a definition of a
derivative and there's multiple ways of
writing this for the sake of this video
I'll write it as the derivative of our
function at Point C this is Lagrangian
with this F prime the derivative of our
function f at C is going to be equal to
the limit as X approaches C of f of X
minus f of C over X minus C and at first
when you see this formula we've seen it
before it looks a little bit strange but
all it is is it's calculating the slope
this is our change in the value of our
function or you could think of it as our
change in Y if Y is equal to f of X and
this is our change in X and we're just
trying to see well what what is that
slope as X gets closer and closer to C
as our change in X gets closer and
closer to zero and we talk about that in
other videos so I'm now going to make a
few claims in this video and I'm not
going to prove them rigorously there's
another video that will go a little bit
more into the proof direction but this
is more to get an intuition and so the
first claim that I'm going to make is if
f is differentiable at x equals C at x
equals C then F is continuous at x
equals C so I'm saying if we know it's
differentiable if we can find this limit
if we can find this derivative at x
equals C then our function is also
continuous at x equals C it doesn't
necessarily mean the other way around
and actually we'll look at a case where
it's not necessarily the case the other
way around that if you're continuous
then you're definitely differentiable
but another way to interpret what I just
wrote down is if you are not continuous
then you definitely will not be
differentiable if s not continuous
at x equals C then f is not
differentiable differentiable at X is
equal to C so let me give a few examples
of a non continuous function and then
think about would we be able to find
this limit so the first is where you
have a discontinuity our function is
defined at C it's equal to this value
but you can see as X becomes larger than
C it just jumps down and shifts right
over here so what would happen if you
are trying to find this limit well
remember all this is is the slope of a
line between when X is some arbitrary
value let's say it's out here so that
would be X this would be the point X
comma f of X and then this is the point
C comma F of C right over here so this
is C comma F of C so if you find the
left-sided limit right over here you're
essentially saying okay let's find this
slope and then let me get a little bit
closer and let me put let's get X a
little bit closer and then let's find
the slope and then let's get X even
closer than that and find this slope and
in all of those cases it would be 0 the
slope is 0 so one way to think about it
the derivative or this limit as we
approach from the left seems to be
approaching zero but what about if we
were take X's to the right so instead of
our exes being there what if we were to
take x's right over here well for this
point X comma f of X our slope if we
take f of X minus f of C over X minus C
that would be the slope of this line if
we get X to be even closer let's say
right over here then this would be the
slope of this line if we get even closer
then this expression would be the slope
of this line and so as we get closer and
closer to X being equal to C
we see that our slope is actually
approaching negative infinity and most
importantly it's approaching a very
different value from the right this
expression is approaching a very
different value from the right as it is
from the left and so in this case this
this limit up here won't exist so we can
clearly say this is not differentiable
so once again not a proof here I'm just
getting an intuition for if something
isn't continuous it's pretty clear at
least in this case that it's not going
to be differentiable let's look at
another case let's look at a case where
we have what sometimes called a
removable discontinuity or a point
discontinuity so once again let's say
we're approaching from the left this is
X this is the point X comma f of X now
what's interesting is where as this
expression is the slope of the line
connecting X comma f of X and C comma f
of C which is this point not that point
remember we have this removable
discontinuity right over here and so
this would be this expression is
calculating the slope of that line and
then if X gets even closer to C well
then we're going to be calculating the
slope of that line if X gets even closer
to C we're going to be calculating the
slope of that line and so as we approach
from the left as X approaches C from the
left we actually have a situation where
this expression right over here is going
to approach negative infinity and if we
approach from the right if we approach
with X is larger than C well this is our
X comma f of X so we have a positive
slope and then as we get closer it gets
more positive more positive approaches
positive infinity but either way it's
not approaching a finite value and one
side is approaching positive infinity or
the other side's approaching negative
infinity this the limit of this
expression is not going to exist so once
again I'm not doing a rigorous proof
here but try to construct a
discontinuous function where you will be
able to find this it is very very hard
and you might say well what about the
situations where s does not even defined
at C which for sure you're not going to
be continuous if f is not defined at C
well if F is not defined at C then this
part of the expression
wouldn't even make sense so you
definitely wouldn't be differentiable
but now let's ask another thing I've
just given you good arguments for when
you're not continuous you're not going
to be differentiable but can we make us
another claim that if you are continuous
then you definitely will be
differentiable well it turns out that
there are for sure many functions an
infinite number of functions that can be
continuous at sea but not differentiable
so for example this could be an absolute
value function it doesn't have to be an
absolute value function but this could
be Y is equal to the absolute value of X
minus C and why is this one not
differentiable at C well think about
what's happening think about this
expression remember this expression all
it's doing is calculating the slope
between the point X comma f of X and the
point C comma f of C so if X is say out
here this is X comma f of X it's going
to be calculate and as we're so as we
take the limit for as X approaches C
from the left we'll be looking at this
slope and then as we get closer we'll be
looking at this slope which is actually
going to be the same in this case it
would be a negative 1 so as X approaches
C from the left this expression would be
negative 1 but as we as X approaches C
from the right this expression is going
to be 1 the slope of the line that
connects these points is 1 the slope on
the line that connects these points is 1
so our the limit of this expression or I
would say the value of this expression
is approaching two different values as X
approaches C from the left to the right
from the left it's approaching negative
1 or it has constantly negative 1 and
until it's approaching negative 1 you
could say and from the right it's 1 and
it's approaching 1 the entire time and
so we know if you're approaching two
different values from on the left-sided
or the right-sided limit then this limit
will not exist so here this is not not
differentiable and even intuitively we
think of the derivative as the slope of
the tangent line and you could actually
draw an infinite number of tangent lines
here's one way to think about it you
could say well maybe this is the tangent
line right over there but why can't I
make something like
this the tangent line that only
intersects at the point C comma zero and
then you could keep doing things like
that why can't that be the tangent line
and you could go on and on and on so the
big takeaways here at least intuitively
and then in a future video I'm going to
prove to you that if F is differentiable
at C then it is continuous at C which
can also be interpreted in that if
you're not continuous at C then you're
not going to be differentiable these two
examples will hopefully give you some
intuition for that but it's not the case
that if something is continuous that it
has to be differentiable it oftentimes
will be differentiable but it doesn't
have to be differentiable and this
absolute value function is an example of
a continuous function at C but it is not
differentiable at C
About this Video
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Dec 22, 2024 Last updated |
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