Learning Outcomes
- Find the derivative of an implicit function by using implicit differentiation.
- 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
-
Use implicit differentiation to find $\dfrac{dy}{dx}$ of the following functions:
(a). $6x^2-2xy=y^2$ (b). $(xy)^2+3y^2=2$ - 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
- $\rhd$ Implicit differentiation (8:01) A discussion on the implicit derivative of the unit circle $x^2+y^2=1$.
- $\rhd$ An example (4:55) Implicit differentiation of $(x-y)^2 = x+y-1$.
- $\rhd$ 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.
- * Practice: Find $y’$ implicitly. (4 problems)
