ANote  dy
dx  is positive if y increases as x increases and is negative if y decreases
as x increases Example 4 The length x of a rectangle is decreasing at the rate of 3 cm/minute and
the width y is increasing at the rate of 2cm/minute When x =10cm and y = 6cm, find
the rates of change of (a) the perimeter and (b) the area of the rectangle Solution Since the length x is decreasing and the width y is increasing with respect to
time, we have
3 cm/min
dx
dt = −
and
2 cm/min
dy
dt =
(a)
The perimeter P of a rectangle is given by
P = 2 (x + y)
Therefore
dP
dt  = 2
2
3
2
2
dx
dt
dy
+dt


 =
− +
= −
(
)
cm/min
(b)
The area A of the rectangle is given by
A = x