DERIVATIVES AND RELATED RATES
LEARNING GOALS
 Understanding the definition of related rates
 Understanding the fundamentals of utilizing related rates
 Able to solve questions with related rates
WHAT IS RELATED RATES ?
Finding a rate at which a quantity changes by relating that quantity to other
quantities whose rates of change are known.
FUNDAMENTALS OF RELATED RATES ?
Fundamentally, if a function F is defined that F = f(x), and x is a function of t.
Then F = f(g(t)), so the derivative of the function F is
𝑭′ 𝒕 = 𝒇′ 𝒈 𝒕
⋅ 𝒈′ 𝒕
Which can be written as:
ⅆ𝑭 ⅆ𝒇 ⅆ𝒙
=
⋅
ⅆ𝒕 ⅆ𝒙 ⅆ𝒕
FUNDAMENTALS OF RELATED RATES ?
The application of chain rule can be extended with the sum, difference, product
and quotient rule of calculus.
For example, if 𝑭 𝒙 = 𝑮 𝒚 + 𝑯 𝒛 then
ⅆ𝑭 ⅆ𝒙 ⅆ𝑮 ⅆ𝒚 ⅆ𝑯 ⅆ𝒛
⋅
=
⋅
+
⋅
ⅆ𝒙 ⅆ𝒕 ⅆ𝒚 ⅆ𝒕 ⅆ𝒛 ⅆ𝒕
Question I: changing rate of water ripples
A stone is dropped into a pond sending out circular ripples moving outward at
0.5m/sec. How fast is the area enclosed by the ripples changing 10sec later?
Practice I
The volume of a spherical balloon is increasing at a constant rate of 50𝑐𝑚3
per second. Find the rate of increase of the radius when the radius is 10cm.
4
[Volume of a sphere = 𝜋𝑟 3 ]
3
Question II: sliding ladder
A 10m long ladder rests against a vertical wall. The ladder is slipping down the
wall, and at the instant when the foot of the ladder is 6m from the wall, it is
moving at 4m/s. At what speed is the top of the ladder moving at this instant?
Practice II
Car A is traveling west at 50km/h and Car B is traveling north at 60km/h. Both are
headed for the intersection of the two roads. At what rate are the cars approaching
each other when Car A is 0.3km and Car B is 0.4km from the intersection?