Learning Outcomes

  1. Find the derivative of an implicit function by using implicit differentiation.
  2. Use implicit differentiation to determine the equation of a tangent line to an implicit function.

Textbook

  • Chapter 3.8  Implicit Differentiation  

Textbook Assignment

  • p. 317:    300-303 all, 309, 311, 315, 319

WeBWork Assignment

  • Derivatives-Implicit

Exit problems of the session 

  1. Use implicit differentiation to find  \dfrac{dy}{dx}  of the following functions:

    (a).  6x^2-2xy=y^2    (b).   (xy)^2+3y^2=2  
  2. Find the equation of the tangent line to the graph of the given equation at the indicated point
    xy+\sin(\frac{1}{2}x)=1,   (\pi, 0).

 

Key Concepts

  • Use implicit differentiation to find derivatives of implicitly defined functions.
  • An implicit defined function is a function that the relationship between the function y and the variable x is expressed by an equation where y is not expressed entirely in terms of x.
  • Implicit differentiation is a technique for computing  \dfrac{dy}{dx}  for a function defined by an equation, accomplished by differentiating both sides of the equation (remember to treat the variable y as a function) and solving for  \dfrac{dy}{dx}

 

Videos and Practice Problems of Selected Topics

  1. \rhd  Implicit differentiation  (8:01) A discussion on the implicit derivative of the unit circle x^2+y^2=1.
  2. \rhd An example (4:55) Implicit differentiation of (x-y)^2 = x+y-1.
  3. \rhd An example (5:24) Find the slope of the tangent line to the curve x^2 +(y-x)^3 = 28 at x=1 using implicit differentiation.
  4. * Practice: Find y' implicitly. (4 problems)

STEM Applications