Implicit Differentiation (DP IB Analysis & Approaches (AA)): Revision Note

Implicit Differentiation

What is implicit differentiation?

• An equation connecting x and y is not always easy to write explicitly in the form or

  • In such cases the equation is written implicitly

    • as a function of and

    • in the form

• Such equations can be differentiated implicitly using the chain rule

• A shortcut way of thinking about this is that ‘ is a function of a ’

  • when differentiating a function of  chain rule says “differentiate with respect to , then multiply by the derivative of ” (which is )

Applications of Implicit Differentiation

What type of problems could involve implicit differentiation?

• Broadly speaking there are three types of problem that could involve implicit differentiation

  • algebraic problems involving graphs, derivatives, tangents, normals, etc

    • where it is not practical to write  explicitly in terms of 

    • usually in such cases, will be in terms of  and 

  • optimisation problems that involve time derivatives

    • more than one variable may be involved too

      e.g.  Volume of a cylinder, 

      e.g.  The side length and (so) area of a square increase over time

  • any problem that involves differentiating with respect to an extraneous variable

    • e.g. but the derivative  is required (rather than )

How do I apply implicit differentiation to algebraic problems?

• Algebraic problems revolve around values of the derivative (gradient)

  • if not required to find this value it will either be given or implied

• Particular problems focus on special case tangent values

  • horizontal tangents

    • also referred to as tangents parallel to the -axis

    • this is when 

  • vertical tangents

    • also referred to as tangents parallel to the -axis

    • this is when 

    • In such cases it may appear that but this has no solutions; this occurs when for nearby values of , 

(i.e.  very steep gradients, near vertical)

• Other problems may involve finding equations of (other) tangents and/or normals

• For problems that involve finding the coordinates of points on a curve with a specified gradient the method below can be used
  

  STEP 1
  Differentiate the equation of the curve implicitly
  

  STEP 2
  Substitute the given or implied value of  to create an equation linking and 
  

  STEP 3
  There are now two equations

  • the original equation

  • the linking equation

Solve them simultaneously to find the  and  coordinates as required

How do I apply implicit differentiation to optimisation problems?

• For a single variable use chain rule to differentiate implicitly

  • e.g. A square with side length changing over time, 

• For more than one variable use product rule (and chain rule) to differentiate implicitly

  • e.g. A square-based pyramid with base length and height changing over time, 

• After differentiating implicitly the rest of the question should be similar to any other optimisation problem

  • be aware of phrasing

    • “the rate of change of the height of the pyramid” (over time) is 

  • when finding the location of minimum and maximum problems

    • there is not necessarily a turning point

    • the minimum or maximum could be at the start or end of a given or appropriate interval