文档介绍：Section
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Differentiation positions
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In this section we will consider the relationship between the derivative of position
of two functions and the derivatives of the individual functions posed. We shall
see that the resulting differentiation rule, known as the chain rule, will be useful in a
variety of situations in our later work. The following example will set the stage.
Example Consider a spherical balloon which is being inflated so that its radius is in-
creasing at a rate of 2 centimeters per second. If we let r denote the radius of the balloon
in centimeters, t denote time in seconds, and V denote the volume of the balloon in cubic
centimeters, then we know that r = 2t and
4
V = πr3.
3
Moreover, we can see that, as a function of t,
4 32
V = π(2t)3 = πt3.
3 3
At time t = 5, the rate of change of the radius with respect to time is
dr
= 2 centimeters per second,
dt
t=5
the rate of change of the volume with respect to the radius is
dV
= 4πr2 = 400π centimeters per centimeter,
dr r=10
r=10
and the rate of change of the volume with respect to time is
dV
= 32πt2 = 800π cubic centimeters per second,
dt t=5
t=5
dV
where dr is evaluated at r = 10 since this is the value of r when t = 5. It follows that
dV dV dr
= .
dt dr dt
t=5 r=10 t=5
1
2 Differentiation positions of Functions Section
That is, the overall rate of change of V with respect to t is the product of the rate of change
of V with respect to r and the rate of change of r with respect to t. This is an example
of the chain rule. Viewed in this manner, the chain rule is saying that if V changes 400π
times as fast as r and r changes 2 times as fast as t, then V changes (400π)(2) = 800π
times as fast as t.
Another interesting special case of the chain rule arises with position of two