3.7 Implicit Differentiation

1
3.7 Implicit Differentiation

• What youll 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, n?I)

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 lt 1,
• then the derivative does not exist at x 0.
• Why?

11
Use the Rational Power Rule

• Find dy/dx of
• Find

• Find

12
Homework

• Page 162
• Exercises 24-42 (3n, n?I),
• 45a, 54