2.5 Introduction:
Find dy/dx.
Practice / Preview:
1.
y = cos 2x
2.
y = cos 3 2x
3.
x y = 1
4.
cos (x y) = 1
dy
/dx = ­2 sin 2x
dy
/dx = ­ 6 cos 2 2x . sin 2x
­1
/dx = x2
dy
?!!
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2.5 Implicit Differentiation
terminology:
y = f(x), (ie. y = x3) expresses y explicitly in terms of x. Equations such as x2 + y2 = 1, xy = 1, 4y3 + x2 ­ 7y = 8, expresses y as one or more implicit functions of x. dy
finding from such an equation is a dx
method called implicit differentiation
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Illustrate
procedure!
explicitly
dy x y = 1 Find dx .
implicitly
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Guidelines for Implicit Differentiation
1.
2.
Differentiate both sides of the equation with respect to x.
dy
Collect all terms involving on one side dx
of the equation.
3.
dy
Factor .
dx
4.
dy
Solve for by dividing.
dx
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dy Find dx .
Examples:
1. x y3 + x y ­ x2 = 4
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work
space!
2.
x y = x ­ 2y
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work
space!
3. cos (x y) = y
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p142 #20
9 y2 ­ x2 = 9
a. Find two explicit functions by solving the equation for y in terms of x.
Sketch and label.
b.
c. Differentiate explicitly.
d.
dy Find implicitly and show that = .
d.
c.
dx 8
b.
work
space!
Sketch and label.
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4
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­4
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­5
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5
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c. Differentiate explicitly.
d.
dy Find implicitly and show that = .
d.
c.
dx work
space!
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Assignment
p142 #1-15 odd, 21-31 odd
after doing Example (p142 #20),
assign #17 & #19
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