example: sec 3.7: implicit differentiation. example: in some cases it is possible to solve such an...
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![Page 1: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/1.jpg)
Example:123 xy
' : yFind
Example: 223 xyxy
' : yFind
Sec 3.7: Implicit Differentiation
![Page 2: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/2.jpg)
Example:
123 xy
Example:
223 xyxy
In some cases it is possible to solve such an equation for as an explicit function
In many cases it is difficult (impossible to write y in terms of x)
Fortunately, we don’t need to solve an equation for y in terms of x in order to find the derivative of y . Instead we can use the method of implicit differentiation
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
Sec 3.7: Implicit Differentiation
![Page 3: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/3.jpg)
Remember:y is a function of x and using the Chain Rule,
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
'yydx
d
2ydx
d
Sec 3.7: Implicit Differentiation
Remember:y is a function of x and using the Chain Rule,
5ydx
d
yedx
d
ydx
dsin
'2yy
'5 4 yy
'ye y
'cos yy
![Page 4: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/4.jpg)
Remember:y is a function of x and using the Chain Rule,
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
'yydx
d
'22 yyydx
d
Example:
223 xyxy
' : yFind
Sec 3.7: Implicit Differentiation
![Page 5: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/5.jpg)
Remember:y is a function of x and using the Chain Rule,
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
')(cossin yyydx
d
Example:
xyyx cos)sin( 2 ' : yFind
Sec 3.7: Implicit Differentiation
![Page 6: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/6.jpg)
Remember:y is a function of x and using the Chain Rule,
This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.
'yydx
d
'22 yyydx
d
32
21
y
y
ydx
d
')(cossin yyydx
d
'yeedx
d yy
y
yy
dx
d
2
'
Sec 3.7: Implicit Differentiation
![Page 7: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/7.jpg)
Remember:
In finding y’’ you may use the original equation
Example:
1644 yx '' : yFind
fat circle
Sec 3.7: Implicit Differentiation
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Sec 3.7: Implicit Differentiation
![Page 9: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/9.jpg)
Sec 3.7: Implicit Differentiation
![Page 10: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/10.jpg)
Sec 3.7: Implicit Differentiation
![Page 11: Example: Sec 3.7: Implicit Differentiation. Example: In some cases it is possible to solve such an equation for as an explicit function In many cases](https://reader030.vdocuments.net/reader030/viewer/2022032805/56649ee15503460f94bf1da4/html5/thumbnails/11.jpg)
Sec 3.7: Implicit Differentiation
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Sec 3.7: Implicit Differentiation
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Sec 3.7: Implicit Differentiation
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Sec 3.7: Implicit Differentiation
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Sec 3.7: Implicit Differentiation
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Sec 3.7: Implicit Differentiation