Let's say I run some

type of a factory, and I've studied my operations. And I'm able to figure

out how my cost varies as a function of quantity over

a week, on a weekly period. And so to visualize

that, let me draw it. I could draw this cost function. So this is my cost axis. This right over here

could be my quantity axis. So that's quantity, or q,

let me just call that q. That's my q-axis.

And my function might

look something like this. It seems reasonable to me. Even if I produce nothing,

I still have fixed costs. I have to pay rent

on the factory. I have to probably pay people

even if we produce nothing. And so let's say that fixed

costs in the week is $1,000. And then as my quantity

increases, so do my costs. So if I produce 100

units right over here, then my cost goes up to $1,300. If I produce more than that,

you see my costs increase and they increase at

an ever faster rate. Now, I go into a lot more depth

on things like cost functions in the Economics

playlist, but what I want to think about in the

calculus context is what would the derivative of

this represent? What would the derivative of c

with respect to q, which could also written as c prime of

q, what does that represent? Well, if we think

about it visually, we know that we can think

about the derivative as the slope of

the tangent line.

So, for example,

that's the tangent line when q is equal to 100. So the slope of that tangent

line you could view as c prime, or it is c prime of 100. But what is that

slope telling us? Well, the slope is the

change in our cost divided by the change in our quantity. And it's the slope

of the tangent line. This is what we first

learned in calculus. As we get to smaller and

smaller and smaller changes in quantity, we

essentially take the limit as our change in

quantity approaches 0. That's how we get that

instantaneous change. So one way to think about it

is this is the instantaneous. This is the rate

right on the margin at which is our cost is changing

with respect to quantity. So if I were to produce just

another drop, another atom of whatever I'm

producing, at what rate is my cost going to increase? And the reason why I'm

saying right on the margin is we see that

it's not constant.

If our cost function

were aligned, we would have a constant slope. The tangent line

would essentially be the cost function. But we see it changes

right over here. The incremental

atom to produce here costs less than the incremental

atom right over here. The slope has gone up. And it might make sense. Maybe I'm using some

raw material out there in the world. And as I use more

and more of it, it becomes more and more scarce. And so the market price of

it goes up and up and up. But you might say,

well, why do I even care about the rate at which

my costs are increasing on the margin? Which is why this is

called marginal cost. Well, the reason why

you care about it is you might be trying to figure

out when do I stop producing? Let's say this is orange juice. If I know that next gallon is

going to cost me $5 to produce and I can sell it for $6,

then I'm going to do it. But if that next

gallon, if I'm up here, and I've already

produced a lot, and I'm taking all the oranges

off the market, and now I have to

transport oranges from the other side of the

planet or whatever it might be, and now if that incremental

gallon of oranges or gallon of orange juice costs

me $10 to produce, and I'm not going to be able

to sell it for more than $6, it doesn't make sense for

me to produce it anymore.

So in a calculus context, or

you can say in an economics context, if you can model your

cost as a function of quantity, the derivative of that

is the marginal cost. It's the rate at which

costs are increasing for that incremental unit. And there's other similar ideas. If we modeled our profit

as a function of quantity, if we took the derivative, that

would be our marginal profit. If we modeled revenue, that

would be our marginal revenue. How much does a

function increase as we increase our

input, as we increase our quantity on the margin?.