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UNIT 2 LESSON 9. IMPLICIT D IFFERENTIATION. IMPLICIT DIFFERENTIATION. So far, we have been differentiating expressions of the form y = f ( x ), where y is written explicitly in terms of x. y = ( x – 4) 3 (2 x + 5) 5. y = 2 x 2 + 5 x – 7. - PowerPoint PPT Presentation
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UNIT 2 LESSON 9
IMPLICIT DIFFERENTIATION
2
IMPLICIT DIFFERENTIATION
So far, we have been differentiating expressions of the form y = f(x), where y is written explicitly in terms of x.
It is not always convenient or possible to isolate the y, and in these cases, we must differentiate with respect to x without first isolating y.
This is called implicit differentiation.
y = 2x2 + 5x – 7 y = (x – 4)3(2x + 5)5562
xxy
2y = y2 + 3x3
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IMPLICIT DIFFERENTIATION
We represent the derivative of y as dxdy
The derivative of x is equal to 1; therefore, it is not
necessary to write it as dxdx
That is, if y = x then
xdxdy
dxd
dxdx
dxdy
1dxdy
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If y is a function of x then its derivative is
y2 is a function of y, which in turn is a function of x.
22 2d d dyy y ydx dx dx
12d y
dx
121
2y dy
dx
Using the chain rule:
Find the following derivative with respect to x
IMPLICIT DIFFERENTIATIONdydx
5
xy xy dx
xddxyd
2
21
21
xdxdy 2
1
21
xdxdy
12 dxdyy
xdxdy
21
xdx
dy2
1
ydxdy
21
METHOD I: Rearrange for y
EXAMPLE 1: a) Differentiate y 2 = x
METHOD II: Implicitly
OR
y = - x ½y = x ½
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EXAMPLE 1: b) Find the slopes of the tangent lines on y 2 = x at (4, 2) and (4, -2)
41
421
dxdy
41
421
dxdy
xdxdy
21
xdxdy
21
(4, 2)
(4, -2)
Using the x value of 4 we need the graph to know which slope goes with which point.
14
m
14
m
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EXAMPLE 1 b) Find slopes of the tangents on y 2 = x at (4, 2) and (4, -2)
41
221
dxdy
ydxdy
21
41
221
dxdy
(4, 2)
(4, -2)
Using the y values of -2 and 2 we know which slope goes with which point.
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undefineddxdy
02
1
undefineddxdy
02
1
undefineddxdy
02
1
EXAMPLE 1 c) Find slope of the tangent on y 2 = x at (0, 0)
ydxdy
21
(0, 0)
xdxdy
21
xdxdy
21
Using the x value of 0
Using the y value of 0
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EXAMPLE 2a: x 2 + y 2 = 25 This is not a function, but it would still be nice to be able to find the slope for any tangent line.
In order to graph this on our calculators we have to rearrange and isolate y
y2 = 25 – x2
225y x 225y x OR
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EXAMPLE 2b: x 2 + y 2 = 25
Take derivative of both sides.
2 2 0dyx ydx
2 2dyy xdx
22
dy xdx y
dy xdx y
2 2 1d d dx ydx dx dx
25
We could differentiate each of the above explicitly but it would be more difficult than using implicit differentiation.
225y x 225y x OR
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dy xdx y
EXAMPLE 2 c) Find the Equation of the tangent line at (3, 4)
So at (3, 4)
x2 + y 2 = 25 (3, 4)
Slope = 43
m
34 (3)4
254
b
b
3 254 4
y x
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EXAMPLE 3: Differentiate 2y = y 2 + 3x 2
22 32 xydxdy
dxd
22 32 xdxdy
dxdy
dxd
xydxdy 622
xdxdyy
dxdy 622
xdxdyy
dxdy 622
yx
yx
dxdy
13
226
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Assignment Questions
Do Questions 1-9 on pages 3 & 4