3.6 - Implicit Differentiation (page ) b We have been differentiating functions that are expressed in the form y=f(x). b An equation in this form is said to define y explicitly as a function of x. b The equations on the next slide are not defined explicitly although, the first two may be manipulated to be written as functions in the form y=f(x) the second two equation are not functions.

Implicit Equations

Implicit Equations (page ) b In the original equations 1,2,3 on the previous slide we could say that the equation defines y implicitly as a function(s) of x. b The 4th equation, called the“Folium of Descartes”, could not be written explicitly to define y in terms of x. Folium of DescartesFolium of Descartes b It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation.

Implicit Differentiation b We differentiate both sides of the equation and treat y as a composite function and apply the chain rule when necessary. b This method of obtaining derivatives is called implicit differentiation. b We will use this technique to solve word problems like the one on the next slide involving “Related Rates”

Related Rate Problem (page 220)

Example 1 (page 213)

Example 1 (page 248)

Example 2 (page 214)

Example 3 (page 214)

Example 4 (page 215)

See graph on next slide.

Example 4 (page 215)

See graph on next slide

Example 4 (page 215)

Differentiability of Functions Defined Implicitly (page 216) b When differentiating implicitly, it is assumed that y represents a differentiable function of x. b If function is not differentiable, then the derivative will be meaningless. b We will not be determining whether or not an implicitly defined function is differentiable. “We will leave such matters for more advanced courses.