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sition” of the hare. Let's say that the course was actually a winding track
with hills and bridges and even a vertical loop, and that the function that
we graphed and previously named x(t) actually measures the linear dis-
tance along this winding path, rather than, say, a horizontal position. To
avoid the horizontal connotations associated with the symbol x, let's intro-
duce the variable s, which gives the distance along the track (in furlongs,
of course).
Let's say that we have a function y(s) that describes the altitude of the
track at a given distance. The derivative dy/ds tells us very basic things
about the track at that location. A value of zero means the course is flat at
that location, a positive value means the runners are running uphill, and a
large positive or negative value indicates a location where the track is very
steep.
Now consider the composite function y(s(t)). You should be able to
convince yourself that this tells us the hare's altitude for any given time t.
The derivative dy/dt tells us how fast the hare was moving vertically, at a
given time t. This is very different from dy/ds. How might we calculate
dy/dt? You might be tempted to say that to make this determination, we
simply find out where the hare was on the track at time t, and then the
answer is the slope of the track at this location. In math symbols, you
are saying that the vertical velocity is y (s(t)). But that isn't right. For
example, while the hare was taking a nap (ds/dt = 0), it doesn't matter
what the slope of the track was; since he wasn't moving along it, his vertical
velocity is zero! In fact, at a certain point in the race he turned around and
ran on the track in the wrong direction (ds/dt < 0), so his vertical velocity
dy/dt would be opposite of the track slope dy/ds. And obviously if he
sprints quickly over a place in the track, his vertical velocity will be higher
than if he strolled slowly over that same spot. But likewise, where the track
is flat, it doesn't matter how fast he runs across it, his vertical velocity will
be zero. So we see that the hare's vertical velocity is the product of his
speed (measured parametrically along the track) and the slope of the track
at that point.
This rule is known as the chain rule. It is particularly intuitive when
written in Leibniz notation, because the ds infinitesimals appear to “can-
cel.”
The Chain Rule of Differentiation
dy
dt = dy
ds
dt .
ds
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