Chapter 3.7
Implicit Differentiation
1
In other words, it doesn't give us an "exact formula" to work with
*This happens when equations are not expressed as y = "something"
Also, your graph won't be a function!! (Doesn't pass the vertical line test)
ex: y 2 = x
x 3 + y 3 ­ 9xy = 0
cos y + 1 = 4x
2
Implicit Differentiation Process
1. Differentiate both sides of the equation with respect to x
dy
2. Collect the terms with on one side of the equation
dx
dy
3. Factor out dx
dy
4. Solve for dx
3
ex: Find dy
if y
dx
2
= x
4
dy
ex: Find if 2y ­ x
dx
2
= 7x 5 + 3y
5
5
dy
ex: Find of 2y = x
dx
2
+ sin y
6
dy
ex: Find of x
dx
2
y + 2y = 3xy
7
ASSIGNMENT
p. 162 #1­8
8
9
ex: Find the slope of the tangent line to the circle x 2 + y 2 = 25 at the point (3, ­4)
10
ex: Find the equation of the tangent line of
x 2 + xy + y 2 = 3 @ (1, 1)
11
ASSIGNMENT
p. 162 #9­12
12
ex: Find the tangent and normal to the ellipse x 2 ­ xy + y 2 = 7 at the point (­1, 2)
13
ex: Find the equation of the tangent and normal line to
x 2 + xy ­ y 2 = 4 at (2, 4)
14
Second Order Derivative
d2y or y''
dx 2
Find y'' of x
4
+ y 4 = 16
15
ASSIGNMENT
p. 162 #17­21, 23, 24, 27, 30
16
ASSIGNMENT
Stewart Ch. 3.6 #27, 29, 31a, 33­35
17