Applications of Chain Rule (DP IB Analysis & Approaches (AA)): Revision Note

Related Rates of Change

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 ’

Differentiating Inverse Functions

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