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.


  • 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