If we have a function y equal to f of x
we now know how to find its derivative at
a particular value of x.
So related to each value of x we get a number
which is equal to the slope of the tangent line there.
Therefore can you tell me if the derivative of a function
is also a function?
Yes, the derivative of a function gives us a new function.
In this video we will understand what this means as well
as its implications.
For example, consider
the function y is equal to x squared.
Previously, we'd found that at
a particular value of x let's say x1,
the derivative of this function is equal to 2 times x1.
But now notice that x1 is an
arbitrary value of the variable x.
That is it can be any value of x.
So let's replace it by x here.
And also let's say the derivative is
denoted by a variable y Prime.
Therefore this means that for each value of x
there is the value y prime related to it
which is the derivative of f of x at that point.
It means this relation is a function.
So we see that finding the derivative of a function
gives us a new function.
This function is called the derived function.
Here the independent variable is x
while the dependent variable is y Prime.
At a particular value of x
the value of the derived function f prime of x
tells us the derivative of f of x at that point.
Now, let me ask you a question.
What will be the domain of the derived function ?
Do you remember
what we mean by domain of a function ?
It's the set of all the values of the independent variable x
for which the function is defined.
So for which values of x
will the derived function not be defined?
Let's say at a particular value of x say x2,
this function is not defined.
Then obviously we cannot find
the derivative of this function at x2.
So it means that this derived function
will not be defined at x2.
So if set 'a' represents the domain of this function
then the derived function
can only be defined for the values of x in set 'a'.
But, now in one of our previous videos
we saw that the derivative does not exist
even at some values of x in the domain of the function.
It's because this limit does not exist.
So this means that at the values of x in set 'a'
where the derivatives does not exist,
the derived function will not be defined.
Therefore, the domain of the derived function is equal to
all the values of x in the domain of this function,
at which the derivative
exists.
Let's take an example to understand this.
Consider this function we saw previously.
We've seen that for this function
even though the function is defined at x equal to zero
the derivative at x equal to zero does not exist.
Finding the derivative at other values of x is simple.
For x greater than zero we will get the derivative
to be constant at a value of 1.
And for x less than 0
we will get the derivative to be constant
at a value of negative 1.
Its graph will look like this.
So we see that for this derived function x equal to 0
will not be in its domain.
But now notice that f prime is a function.
So can we differentiate this derived function too ?
What does this mean? Let's find this out in the next part
This tells us
that the variable y prime is a function of the variable x.
So it means that as
the value of x changes, the value of y prime changes.
So we can find the instantaneous
rate of change of y Prime with respect to x,
that is the derivative of f prime of x
Similar to this, it will be equal to the limit
of the average rate of change of f prime of x.
Now since this is a derivative of a derived function of
f of x, it's called the second derivative of f of x.
So it's denoted by putting double Prime on f like this.
Now as seen earlier,
the differentiation of f prime of x gives us
a new function called the second derived function.
So we can further differentiate it.
This gives us the third derivative of f of X.
Similarly, we can find the fourth derivative and so on...
These are all called the higher-order derivatives of f of x.
But what do these higher-order derivatives tell us?
Do they have any significance?
Well, as we look at advanced topics of calculus , we will
understand why the higher-order derivatives are
significant
For now, let's look at an example and understand
the importance of the second derivative
Consider this object in motion
We note that the speed of the object
tells us how fast or slow it changes its position
That is the rate of change in position
with respect to time
But it's the speed of any object in motion
always constant.
No, right
For example as the object starts from rest
its speed starts increasing from zero
We say that it accelerates
And when the object has to come to rest
its speed decreases and we say that it decelerates
So acceleration of an object
is the rate of change in speed with respect to time
And we know that speed
is the rate of change in position with respect to time.
So we see that acceleration tells us
the second-order change in the position of the object
Let's see what we mean by it
Let's say the object's position is given by
this function of time
We have seen previously that the instantaneous
speed of the object at a time t1
Will be equal to the derivative of this function at t1
So the derivative of this position function
will give us a new function
This function will tell us the speed of the object at a
time (t)
Now, can you tell me how we can find the
instantaneous acceleration at time t1?
It will be given by derivative of the speed function at t1,
right
This means the acceleration function will be given by the
second derivative of the position function
So we know that the second derivative of
a position function tells us about
one important physical quantity
But wait, there is something more fascinating
It turns out that for any object in motion
the force acting on it is equal to its
mass times acceleration
We know that the mass of an object is constant
So we see that the second derivative of the
position function
Tells us the force acting on the object
Therefore while studying the motion of an object
Finding the second derivative of the position function
plays a central role
So until now we've seen many aspects of
the derivative of a function
Also, we've seen related to this idea
that there are many notations involved
One particularly important one we haven't talked about is
this one
dy by DX or
DF by DX
This notation suggests that the derivative of a
function is a ratio of two quantities
But we know this is not the case as we've seen before
the derivative is the result of the limit of this ratio
It's not a ratio in itself
But we will see how useful this notation is
as we delve deeper into calculus
In essence it captures the whole idea of the derivative
let's see how
Consider this function
this symbol Delta is always used to denote the
change in a quantity
so if the value of x changes from x1 to x2
Then Delta x meets x2 minus x1
In other words x2 will be equal to x1 plus Delta x
Now let's say at x1.
The value of the function is y 1
and at x 2 the value of the function is y 2
So Delta y means y2 minus. y1
Therefore the numerator here can be denoted as Delta y
Now Delta x tends to 0 means
the change in x will get very very small close to 0
We say that the change is
infinitesimal
We denote this infinitesimal change in x by dx
Now we see that as Delta x tends to 0
Delta y will also get smaller and smaller
so we say that the change in y is also
Infinitesimal and we denote it by dy
Now since the derivative
is the limit of the ratio change in y over change in x
We can intuitively think of it
as the ratio of infinitesimal change in y
over infinitesimal change in x
It means the derivative tells us how the
value of y changes for an infinitesimal change in X
Hence we can see that this notation helps us intuitively
understand the derivative of a function
But always be aware that the derivative of a function
is not a ratio of two quantities
It's a number which is a limit of the ratio of the
two quantities
Would you rather have any other notations
for the derivative of a function
Share your thoughts in the comment section below
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