Problem :
Suppose there is a 10 foot ladder leaning against a wall, the base of which is being
pulled away from the wall, along the ground, at a constant rate of 1 foot per second.
The top of the ladder remains in contact with the wall as the base moves. How quickly is
the top of the ladder sliding down the wall when it is 5 feet from the ground?
Let
B(t) be the distance of the base of the ladder from the wall and let
T(t) be the
distance of the top of the ladder from the ground. These functions satisfy the relation
g(t) = . 

Differentiating each side with respect to
t, we have
g'(t) = w'(t) 

We are given that
g'(t) = 1 and are interested in the situation when
w(t) = 5. Solving
for
w'(t) above and plugging in these values, we find that the top of the ladder has
velocity
or approximately
1.73 feet per second downward. It is intriguing to note that as the
top of the ladder approaches the ground, its speed approaches infinity, even though the
bottom of the ladder continues to move away at a constant rate! (Realistically, at some
point the bottom of the ladder will slip, the top crashing to the ground quite suddenly.)
Problem :
Suppose you are given a magic rectangle, which can be stretched vertically or horizontally
to change the lengths of its sides, but such that the area remains constant. You are given
the rectangle in the form of a square, with each side have length 1 foot. To make sure
the rectangle really is magic, you pull on it in one direction so that two opposite sides
increase in length at a rate of 3 inches per second. Sure enough, the other two sides of
the rectangle shrink to maintain the area of 1 square foot. How quickly are they
shrinking when they are half their original length?
We choose to work in inches. Let
a(t) be the length of the sides that are expanding at
time
t and
b(t) the length of the sides that are shrinking. Then
a(t)b(t) = 144.
Solving for
a(t) and differentiating each side with respect to
t gives
a'(t) = b'(t) 

We are given that
a'(t) = 3 and are interested in the moment when
b(t) = 6. Solving for
b'(t) and plugging in these values, we obtain
Thus the sides are shrinking at
3/4 inches per second when they are at half their
original length.
Problem :
Suppose a point is moving along the curve y = 3x^{2}  2x from left to right at a horizontal
speed of 2 units per second. How quickly is the ycoordinate of the point changing
when the xcoordinate is at 1?
We differentiate each side of
y = 3x^{2}  2x with respect to
t:
Substituting
x'(t) = 2 and
x(t) =  1, we obtain
y'(t) =  16.