Objectives: In this tutorial, we define what it means for a realtion to define a function implicitly and give an example. Then, using several examples, we demonstrate implicit differentiation which is a method for finding the derivative of a function defined implicitly. After working through these materials, the student should be able

• to find the derivative of an implicitly defined function using implicit differentiation;
• to find the equation of a tangent line to a curve which implicitly defines a function.
Modules:

 Definition. A relation F(x,y) = 0 is said to define the function y = f(x) implicitly if, for x in the domain of f, F(x,f(x)) = 0.

• Discussion [Using Flash]

 Comment. Given a differentiable relation F(x,y) = 0 which defines the differentiable function y = f(x), it is usually possible to find the derivative f' even in the case when you cannot symbolically find f. The method of finding the derivative which is illustrated in the following examples is called implicit differentiation.

• Examples

1. Discussion [Using Flash]

2. Discussion [Using Flash]

3. Discussion [Using Flash]

4. Discussion [Using Flash]

5. Discussion [Using Flash]

6. Discussion [Using Flash]

• Drill problems on finding the equation of the tangent line using implicit differentiation.

• Drill problems on finding the derivative using implicit differentiation.

• A LiveMath notebook illustrating the method of finding the derivative using implicit differentiation.