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

    • e.g. 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 ’

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

How do I find a derivative using implicit differentiation?

• In order to find for an implicit relation between and , follow these steps

• STEP 1
  Differentiate both sides of the equation implicitly with respect to

  • E.g.  

• STEP 2
  Rearrange the new equation to make  the subject

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 (therefore) 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)

• 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 

  • Recall the relation

• 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, 

    • Differentiating both sides with respect to gives

• 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, 

    • Differentiating both sides with respect to gives

• 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

    • The solution is not necessarily at a turning point

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