Related Rates of Change

Example 1:

The variables and are related by the equation .

Given that is increasing at the rate of units per second when , find the rate at which is increasing at this instant.

Step 1: Identify the components we want, got, and need.

Want: the rate at which is increasing.

Got: the rate at which is increasing.

want got need

need

To find the "need" we have to balance the right side which can only be done with .

Step 2: Find by applying the product rule.

First, can be rewritten as . This is a power function. The angle is what’s inside the bracket, which is in this case. The angle differentiates to . Then, for the power function, we bring down the power, knock off the power, then the angle inside stays the same.

Angle stays the same



Differentiated angleDifferentiated power function

So, using the product rule,

Step 3: Find the value of when .

Step 4: Substitute the values we obtained to find .

want got need

Therefore, the rate of increase of is units per second.

Example 2:

The volume of a cube is decreasing at a constant rate of cms.

a) Find the rate at which the length of one side of the cube is decreasing when the volume is cm, giving your answer to significant figures.

Step 1: Identify the components we want, got, and need.

Want: the rate at which is increasing.

Got: the rate at which the volume is increasing (by a negative value i.e. decreasing).

want got need

Step 2: Find when .

We did not get given an equation so we must construct one ourselves.

The question is asking about the volume of a cube. The formula for the volume of a cube may be given by , where we let the side length be .

Differentiating with respect to gives

To find , we reciprocate , giving

When , .

So,

Step 3: Substitute the values we obtained to find .

want got need

Reminder: In general, means the rate of increase of . When it’s negative, then it signifies that is instead decreasing.

Therefore, the rate at which the length of one side of the cube is decreasing is approximately cms, correct to significant figures.

b) Find the volume of the cube when the length of one side is decreasing at the rate of
mms.

Step 1: Find the side length when one side is decreasing at the rate of mms.

We rewrite in the correct unit to start with. We know that to convert from mm to cm, we divide by .

mms cms

Multiplying both sides by ,

Dividing both sides by ,

Step 2: Find the volume when .

cm

Example 3:

The variables , , and are related by the equations:

and,

Find the value of , when .

Leave your answer correct to significant figures.

Step 1: Identify the components we want, got, and need.

Want: the rate at which is increasing.

Got: the rate at which is increasing.

want got need

This is a unique question in that they haven't given us a rate of change, but instead two equations in three variables.

Therefore, we need to find both and .

Step 2: Differentiate both the equation with respect to and the equation with respect to to get and , respectively.

is a power function and can be rewritten as . The angle is what’s inside the bracket, which is in this case. The angle differentiates to . Then, for the power function, we bring down the power, knock off the power, and then the angle inside stays the same.

Angle stays the same



Differentiated angleDifferentiated power function

is an exponential function. The angle is what’s in the power, which is in this case. The angle differentiates to . Then, the function of stays the same in differentiation as well as its power. The is a multiplier so we can write that down first.

MultiplierAngle stays the same



Differentiated angle

Step 3: Find and when .

Notice how the question gives us , but our formula is in terms of . We substitute into our equation to find the value of .

When ,

We have , so we substitute that into .

Step 4: Find when .

Step 5: Substitute the values we obtained to find .

want got need

To significant figures, .