The Derivative: Interpretations, Notations, and Units
Goal: Students get used to Liebniz notation as well as the prime notation, and understand that
derivatives are rates, so units are appropriate
We are now comfortable with the concepts of instantaneous and average rate of change. The
average rate is Δy / Δx, the amount of change of y divided by the amount of change of x. The
instantaneous rate is the limit of this as Δx goes to zero. This suggests a number of things.
One, is that derivatives should have units. At least, if we are measuring real life things, real life
units should be attached to our measurements! We started here by thinking about derivatives in
terms of velocity versus position. The change in position is a distance, and the change in time is
measure in seconds (or hours, or whatever), so the average velocity of a car might be measured
in miles per hour. But any units will work just as well:
Example: A mountain climber noted that the temperature at his base camp at 8000 feet was
38°F. After climbing for a while, he was at 12000 feet and the temperature had dropped to 28°F.
We measure the average rate of change: ΔT / Δh = –10°F / 4000 feet. So the temperature is
dropping (hence the negative sign) at a rate of 2.5° per 1000 feet.
Exampel: (Hughes-Hallett p.95 example 5) if P = f(t) is the population of Mexico (in millions)
where t is the number of years since 1980, interpret:
a) f′(6) = 2: f′ has units of ΔP / Δt = millions of people per year, so if the population growth rate
remains unchanged, then the population would grow by two million people per year in 1986.
b) f-1(95.5) = 16: this is the inverse, so it turns millions back into year. In this case, it tells us that
the population hit 95.5 million 16 years after 1980 (so in 1996).
c) f-1′(95.5) = 0.46: look at units! The output of f-1 is in years while its input is in millions of
population. So we are looking at years/million, in other words what we are being told is that
when the population is 95.5 million, it takes about 0.46 years to add another million.
Example: many cars get reduced gas mileage if you drive faster. Let’s say you have a car that
gets 30 mpg when it is travelling 60 mph, but for each mpg faster you drive, it loses 0.3 mpg.
We can think of a function m(s) that measures mileage for a given speed. For instance, we know
that m(60) = 30, while m(70) = 27. We also know that m′(60) = –0.3. What does that mean?
Well, the units are (miles/gallon) / (miles/hour) = hour/gallon. So it means that if we are willing
to use an extra gallon of gas we can take 0.3 hours off our trip by going faster.
Now the f′ notation was invented by Newton, because he was focused on the functions. Liebniz,
who independently co-discovered calculus was focused on the changes, the Δy / Δx as Δx went to
zero. To emphasize these relative rates of change he used a different notation: dy / dx. The idea
is that as Δx gets smaller and smaller, it becomes the “infinitesimally small” dx and causes only
an “infinitesimal change” in y, dy, and we can take the ratio of these to find the infinitesimal (or
instantaneous) rate of change dy / dx. Now the idea of infinitesimal change didn’t stick, but the
notation did. (Well, there are theories of calculus that use infinitesimals, and they are easier in
some ways and harder in others, but they never really caught on, so we don’t use them until we
get to a much more complicated theory of calculus on curved surfaces where they are more
convenient than the tradition approach.) So we often speak of the derivative, dy / dx or df / dx.
But keep in mind that the dy is a single object, which cannot be broken down, so we can’t
“cancel the d’s” for instance.
The more important idea is that we think of d / dx as an instruction to “differentiate with respect
d
df
to x.” So if you have a function f(x) we might take
which means to find the derivative
f=
dx
dx
of f with respect to x. This notation is especially useful if there are several variables flying
around and you need to know which is the independent one. For instance, the amount of money
in an investment might depend on the interest rate, the duration of the investment, and the initial
investment. So you might have a function V(r, t, P) where V is the value, r the interest rate, t the
time since investing began, and P the principal. Once you’ve put in P and locked in interest rate
r, you are probably only interested in how much you earn per year, so you want to know about
dV/dt. But if you are comparing investments with different interest rates you might be more
curious about dV/dr.
The downside of the new notation is that it is inconvenient to talk about evaluating the
df
derivative. Since we are focusing on change, something like
(3) doesn’t make sense, because
dx
df
if x is 3 it is not changing. We instead use the more cumbersome notation
which reads
dx x=3
“the derivative of f with respect to x, evaluated when x = 3.”
Problems: (Hughes-Hallett) 2.4: 2, 5, 9, 12, 13, 17, 18, 26