1. DO NOW: Find the inverse of: How do you find the equation of a tangent line? How do you find points where the tangent line is horizontal?

2. HW: Pg. 162 #2-20e ASWDTAT: Understand the difference between implicit and explicit functions, and differentiate implicitly. 3.7 - Implicit Differentiation

3. Introduction Sometimes an equation only implicitly defines y as a function (or functions) of x. Examples x 2 + y 2 = 25 <- we will solve for y on next slide x 3 + y 3 = 6xy <- very difficult to solve for y

4. Introduction Solving for y actually gives two functions

5. Implicit vs. Explicit functions

6. Implicit functions A function whose relation to the variable is given by an equation for which the function has not been solved explicitly. For example, in the equation x 2 + y 2 = 1, y is an implicit function of x.

7. Implicit Differentiation There is a way to find the derivative of these functions called implicit differentiation to find dy/dx without solving for y: 1. First differentiate both sides of the equation with respect to x 2. Then solve the resulting equation for y’

8. Example 1 For the circle x 2 + y 2 = 25, find (a) dy/dx using both implicit differentiation and by first solving for y (b) An equation for the tangent at the point (3, 4)

9. Example 2 For x 3 + y 3 = 6xy (a) Find y’ (b) Find the tangent to the curve at the point (3,3) (c) At what points on the curve is the tangent line horizontal?

10. Example 2 - Solutions

11. Example 3 Find y’ if sin(x + y) = y 2 cosx

12. Example 4 Find the tangent line and normal line on at the point (1,1) How do we graph it on the calculator?

13. Solution

14. Example 5 The equation xy = c, c≠ 0 represents a family of ____________. x 2 – y 2 = k, k≠0 represents another family of _____________, with asymptotes y = _____. Show that the families are orthogonal trajectories of each other.

15. Example 4 (solution)