When we first started to talk about derivatives, we said that becomes

when the change in x and change in y become very small.

dy can be considered a very small change in y.

dx can be considered a very small change in x.

Differentials∆ 𝑦∆𝑥

𝑑𝑦𝑑𝑥

Let be a differentiable function.

The differential is an independent variable.

The differential is:

y f xdxdy

dy f x dx

dy = 𝑑𝑦𝑑𝑥

Related RatesIf we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. However, it is much easier to measure directly the rate of increase of the volume than the rate of increase of the radius.

In this section, we will learn:How to compute the rate of change of one quantity

in terms of that of another quantity.

In a related-rates problem, the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity—which may be more easily measured. The procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time.

Example 1: Consider a sphere of radius 10cm.

If the radius changes 0.1cm (a very small amount) how much does the volume change?

34

3V r

24dV r dr

24 10cm 0.1cmdV

340 cmdV

The volume would change by approximately 340 cm

𝑑𝑟=0.1𝑐𝑚𝑑𝑉=?

Example 2: Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec. At what rate is the sphere growing when r = 10 cm?

34

3V r

24dV dr

rdt dt

2 cm4 10cm 0.1

sec

dV

dt

3cm40

sec

dV

dt

The sphere is growing at a rate of 340 cm / sec

Example 3: Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?

34

3V r

2

1 1100

4 (25) 25

dr

dt The radius of the balloon is increasing

at the rate of 1/(25π) ≈ 0.0127 cm/s.

L3

sec

dV

dt

3cm3000

sec

dh

dt

2V r h

2dV dhr

dt dt

3

2

cm3000

secdh

dt r

Example 4: Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?

= ?

r is constant

Steps for Related Rates Problems:

1. Draw a picture (sketch).

2. Write down known information.

3. Write down what you are looking for.

4. Write an equation to relate the variables.

5. Differentiate both sides with respect to t.

6. Evaluate.

Example 4: Hot Air Balloon Problem

@4

rad0.14

min

d

dt

How fast is the balloon rising?

tan500

h

2 1sec

500

d dh

dt dt

2

1sec 0.14

4 500

dh

dt

h

500ft

2

2 0.14 500dh

dt

Example 5: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.

How fast is the distance between the trucks changing 6 minutes later?

𝑑=𝑣𝑡

z2 = x2 + y2 @ 6 min z = 5 mi

@ 6 min x= 4 mi & y = 3 mi from initial position

2 𝑧𝑑𝑧𝑑𝑡

=2 𝑥𝑑𝑥𝑑𝑡

+2 𝑦𝑑𝑦𝑑𝑡

5𝑑𝑧𝑑𝑡

=4 ∙40+3 ∙30

𝑑𝑧𝑑𝑡

=50𝑚𝑖 /h𝑟

Example 6: Car A is traveling west at 50 mi/h and car B is traveling north at 60 mi/h. Both are headed for the intersection of the two roads.

At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection?

dx / dt = –50 mi/h & dy / dt = –60 mi/h. both x & y are decreasing

𝑑𝑧𝑑𝑡

=?

z2 = x2 + y2 ⇒ 2 𝑧 𝑑𝑧𝑑𝑡 =2 𝑥𝑑𝑥𝑑𝑡

+2 𝑦𝑑𝑦𝑑𝑡

When x = 0.3 mi & y = 0.4 mi, z = 0.5 mi.