We will use the derivative of profit, cost, and revenue functions to make estimates.
- To understand this process, we look first at an example function
What is the rate of change of 
at 
?
We calculate the derivative and then substitute 1 for x:
Graphs illustrating this process are shown below. The tangent
line at
is
We have
as an
equation of the tangent line. Examine the points on the
graph and on the tangent where
. They are not far apart. The tangent
line gives an estimate of the actual
point on the original function
The actual
function value on the
graph is 10. The tangent
line estimates this at 9. This estimate is better the closer we are to the
point of tangency, in this case the
point where
. This estimate is helpful for more complicated functions in arriving quickly at a value which is “close” to the next value of the function.
Notice that the
point on both the graphs of
and the tangent
line at
is (1, 4). The slope, or derivative of
, of the tangent
line is 5 at that
point as we calculated above. This means the
function increased about 5 units from the
point where
to the
point where
. We use this derivative in
marginal analysis.
Let’s Practice:
Marginal profit is the derivative of the profit
function (the same is true for cost and revenue). We use this marginal profit
function to estimate the amount of profit from the “next” item.
For example: using our profit
function from test #1, we have
Therefore,
The marginal profit at
is
P'(300) = 80 + 0.2(300) - (40 + 0.4(300)) = -$20
Using the ideas from the example above, we know that -20 represents an estimate of the change in profit from the points on the profit
graph at
and
. An estimate of the profit from the 301
st item is -$20, meaning that production of the 301
st item will decrease profit by $20.
It is important to be able to compare this to the exact change in profit as we move to the 301
st item. Remember that the derivative gives us a
point on the tangent line, which is not exactly on the actual
graph of the profit function.
The exact profit from the 301st item is given by
P(301) - P(300) = $979.90 - $1000 = -$20.10
We can see that this estimate of -$20 from the derivative of
is very close to the exact value.
Summary:
An estimate of the amount of profit from the 301st item (the change in profit as we go from 300 items to 301 items) is given by
P'(300) = -20
The exact change in profit is given by
P(301) - P(300) = -$20.10