5.8.2 Applications of Chain Rule

What is meant by rates of change?

• A rate of change is a measure of how a quantity is changing with respect to another quantity
• Mathematically rates of change are derivatives
  • could be the rate at which the volume of a sphere changes relative to how its radius is changing
• Context is important when interpreting positive and negative rates of change
  • A positive rate of change would indicate an increase
    • e.g. the change in volume of water as a bathtub fills
  • A negative rate of change would indicate a decrease
    • e.g. the change in volume of water in a leaking bucket

What is meant by related rates of change?

• Related rates of change are connected by a linking variable or parameter
• this is often time, represented by 
• seconds is the standard unit for time but this will depend on context
• e.g.  Water running into a large hemi-spherical bowl
• both the height and volume of water in the bowl are changing with time
• time is the linking parameter between the rate of change of height and the rate of change of volume

How do I solve problems involving related rates of change?

• Use of chain rule and product rule are common in such problems
• Be clear about which variables are representing which quantities

STEP 1
Write down any variables and derivatives involved in the problem
e.g. 

STEP 2
Use an appropriate differentiation rule to set up an equation linking ‘rates of change’
e.g.  Chain rule:   

STEP 3
Substitute in known values
e.g.  If, when ,  and , then 

STEP 4
Solve the problem and interpret the answer in context if required
e.g.    ‘when ,  changes at a rate of 4, with respect to ’

What is meant by an inverse function?

• Some functions are easier to process with  (rather than ) as the subject
• i.e.  in the form
• This is particularly true when dealing with inverse functions
• e.g.  If  the inverse would be written as
• finding can be awkward
• so write  instead

How do I differentiate inverse functions?

• Since it is easier to differentiate “ with respect to ” rather than “ with respect to ”
  • i.e.  find rather than 
  • Note that will be in terms of  but can be substituted

STEP 1
For the function , the inverse will be

Rewrite this as

STEP 2
From find 

STEP 3
Find  using - this will usually be in terms of 

• If an algebraic solution in terms of  is required substitute  for  in
• If a numerical derivative (e.g. a gradient) is required then use the -coordinate
  • If the -coordinate is not given, you should be able to work it out from the orginal function and -coordinate