In this video, we will
answer the question,
why is the derivative
of the area of a circle
equal to the circumference of a circle,
where the area equals pi r-squared
and the circumference equals two pi r.
Beginning with the area
forming the first circle,
let's determine dA dr,
or the derivative of A
with respect to r,
which we can also denote A prime of r.
To find the derivative,
we need to apply the power
rule of differentiation,
shown here and therefore,
the derivative is equal to pi
times the derivative of r-squared
with respect to r,
giving us pi times two r,
raised to the power of one.
Simplifying, we have
dA dr, or A prime of r,
equals two pi r, which is the
circumference of a circle.
So the question is why is this derivative
equal to the circumference?
Let's first review some notation,
differential A or dA describes
the change in the area.
Differential r or dr describes
the change in the radius.
And therefore dA dr or A prime of r,
describes the change in area
with respect to the radius.
And now beginning with
a circle, with radius r,
pictured here in blue,
let's consider a small increase
in the radius of the circle.
Let's say this will increase here.
We can label the change in the radius,
or this thickness differential r
and now the area of the red ring
represents the change in the area,
given differential r, a
small change in the radius.
To determine the change in the area,
we can determine the area of this ring
by cutting the ring and unrolling it
which forms this long rectangle here.
Where the length of the rectangle
is equal to the
circumference of the circle,
which is two pi r,
and the width of the rectangle
equals the change in the
radius or differential r.
And because the area of a rectangle
equals length times width,
the area of the red ring is equal to
two pi r times differential r.
Which again represents
the change in the area
and therefore we have differential A
equals two pi r times differential r.
From this equation, if
we divide both sides
of the equation by differential r,
we have dA dr, which is
equal to the derivative of A,
with respect to r,
equals two pi r, which is the
circumference of the circle.
So the derivative of the area of a circle
is the circumference.
Increasing the radius will result
in an increase in area
proportional to the circumference.
Now, I do want to mention,
this is an informal discussion
as to why the derivative
equals the circumference.
If we go back up to the original circle,
when determining the area of the red ring,
which represents the change in the area,
there's really an inner
radius and an outer radius.
Where the inner radius is r
and the outer radius is
really r plus differential r.
Remember, where differential
r is the change in the radius.
And, therefore, when we cut the ring
and unroll it to form this rectangle here,
one length is two pi r,
but the other length is really
two pi times the quantity r
plus the differential r.
But because differential
r is very, very small,
we are treating the two opposite
sides of the rectangle as
two pi r to determine
the area of the ring.
I hope you found this helpful.
