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Unformatted text preview: 245 3.3. DERIVATIVES In the ﬁrst term above, the ﬁrst factor is ﬁxed and the second goes to zero
as △t → 0, while in the second term, the ﬁrst factor is bounded (since
→
→
△− /△t converges to − ) and the second goes to zero. Thus, the whole
x
v
mess goes to zero, proving that the aﬃne function inside the absolute value
in the numerator on the left above represents the linearization of the
composition, as required.
An important aspect of Proposition 3.3.6 (perhaps the important aspect)
is that the rate of change of a function applied to a moving point depends
only on the gradient of the function and the velocity of the moving point
at the given moment, not on how the motion might be accelerating, etc.
→
For example, consider the distance from a moving point − (t) to the point
p
(1, 2): the distance from (x, y ) to (1, 2) is given by
f (x, y ) = (x − 1)2 + (y − 2)2 with gradient
→
−
∇ f (x, y ) = (x − (x − 1)
1)2 + (y − 1)2 →
−+
ı (x − (y − 2)
1)2 + (y − 1)2 →
−. If at a given moment our point has position
→
− (t ) = (5, −3)
p0
and velocity
→
− (t ) = −2− − 3−
→
→
v0
ı then regardless of acceleration and so on, the rate at which its distance
from (1, 2) is changing is given by
d
dt t=t0 →
−
→
→
[f (− (t))] = ∇ f (5, −3) · − (t0 )
p
v 4−
→ − 3 − · (−2− − 3− )
→
→
→
ı ı 5
5
89
=− +
55
1
=.
5
The other kind of chain rule that can arise is when we compose a
realvalued function f of a vector variable with a realvalued function g of
a real variable:
= ...
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 Fall '08
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 Calculus, Derivative

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