3.7 Implicit Differentiation

1. 3.7 Implicit Differentiation What you’ll learn about… Implicitly defined functions Lenses, tangents, and normal lines Derivatives of Higher Order Rational Powers of Differentiable Functions Why? Implicit differentiation allows us to find derivatives of functions that are not defined or written explicitly as a function of a single variable.

2. Differentiating a Function in Terms of Both x and y Find dy/dx if y2 = x. • Differentiate both sides with respect to x • Get dy/dx on one side and all else on the other. • The graph shown on p157 gives the curve and the tangent lines at the points (4,2) and 4,-2). Dy/dx gives the slope of both of these lines. You try:

3. Find the slope of the circle at the point (2,-2) using example 2. Find dy/dx Put point (x,y) into formula to find slope

4. Show that the slope dy/dx is defined at every point on the graph of 2y = x2 + sin y. HOW? Differentiate both sides with respect to x. Get all dy/dx terms on one side of equation, all else on the other. Factor dy/dx out, group other factors with () Divide to get dy/dx alone

5. Lenses, Tangents, and Normal Lines In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light make with the line perpendicular to the surface of the lens at the point of entry. This line is called the normal to the surface at the point of entry. In a profile view of a lens, the normal is a line perpendicular to the tangent to the profile curve at the point of entry. (p159 / figure 3.51) Profiles of lenses are often described by quadratic curves. When they are, we can use implicit differentiation to find the tangents and normals.

6. Find the tangent and normal to the ellipse x2 – xy + y2 = 7 at the point (-1,2). Differentiate to find dy/dx. Use the product rule to differentiate xy, group terms in ( ). Find slope of tangent using dy/dx. Write the tangent equation using that slope and the point (-1, 2). Write the normal equation using the opposite reciprocal slope and the point (-1, 2).

7. Homework Page 162 Exercises 3-21, (3n, nЄI)

8. Warm Up Page 164 Exercises 59-64 Skip #62 No Calculator!

9. Finding a Second Derivative Implicitly Find if 2x3 – 3y2 = 8. y’ = y”= sub y’ into 2nd derivative and simplify

10. Rule 9 Power Rule for Rational Powers of x If n is any rational number (fraction), then If n < 1, then the derivative does not exist at x = 0. Why?

11. Find dy/dx of Find Find Use the Rational Power Rule

12. Homework Page 162 Exercises 24-42 (3n, nЄI), 45a, 54