U1S5V05 Areas of Circles and Derivatives.txt

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So you just did some problems where
you computed the derivative of the area of a circle
with respect to different variables.
In the first problem, we found a formula
for the area of a circle in terms
of the radius of a circle, and we computed the derivative
with respect to the radius, dA/dr. In the second problem,
we were interested in how the area changed with respect
to the circumference of the circle
because we were building a hot tub,
so we found a formula for the area
in terms of c, the circumference.
And the derivative we were looking at was dA/dc.
Of course, these two derivatives are both functions
and they're not equal because they're
describing different rates with respect to different variables.
In the real world, what variable we want to differentiate
with respect to is determined by the problem
we're trying to solve.
For example, suppose we have a leaky pipe
and it is dripping, causing a circular puddle
to grow beneath it.
Then we're interested in how the area of this puddle
is changing in time, so we're looking at dA/dt.
So in each of these examples, it's
really clear what the quantity is that we're looking at.
We're looking at the area and we're
differentiating with respect to radius, circumference,
and time.
Now if we want to look at these very same examples but using
prime notation, well, first we have to define three entirely
new functions.
We could say that A is equal to f of r,
and then the derivative would be f prime of r.
In this case, we'd have to define some function g
of c that's equal to the area.
Then the derivative we'd be looking at is g prime of c.
In the last example, we could define
a function h of t and the derivative
we're looking at is h prime of t.
So to use prime notation, we had to introduce
three new functions.
And now it's not even clear from the names of these functions
what the units are, and we can't even
tell that these derivatives are rates
of change of areas of circles.
So the derivatives are slightly easier to write,
but we no longer know what they mean.
And if we evaluate it at a point,
then it's even less clear what the units are.
So when you're modeling a problem,
it's really great to use Leibniz's notation.
It really helps you keep track of what the physical quantities
are that you're really interested in and making sure
that you're differentiating them with respect
to the variables of interest.
So why don't you do a couple more problems
to get used to using Leibniz's notation
before we continue learning more about the derivative.