implicit differentiation. implicit differentiation look at the graph of even though this is not a...
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![Page 1: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/1.jpg)
IMPLICIT DIFFERENTIATION
![Page 2: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/2.jpg)
Implicit Differentiation
Look at the graph of
Even though this is not a function the gradient isclearly defined at every point (except at the twopoints on the x-axis where the gradient is infinite)
It is called an implicit function as it does not give y explicitly in terms of x unlike
eg 122 yx
122 yx
![Page 3: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/3.jpg)
Implicit Differentiation
Look at the graph of
Even though this is not a function the gradient isclearly defined at every point (except at the twopoints on the x-axis where the gradient is infinite)
It is called an implicit function as it does not give y explicitly in terms of x unlike
eg 122 yx
122 yx
![Page 4: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/4.jpg)
Implicit Differentiation
Look at the graph of
Even though this is not a function the gradient isclearly defined at every point (except at the twopoints on the x-axis where the gradient is infinite)
It is called an implicit function as it does not give y explicitly in terms of x unlike
eg 122 yx
122 yx
![Page 5: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/5.jpg)
Implicit Differentiation
Look at the graph of
Even though this is not a function the gradient isclearly defined at every point (except at the twopoints on the x-axis where the gradient is infinite)
It is called an implicit function as it does not give y explicitly in terms of x unlike
eg 122 yx
122 yx
![Page 6: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/6.jpg)
Implicit Differentiation
Look at the graph of
Even though this is not a function the gradient isclearly defined at every point (except at the twopoints on the x-axis where the gradient is infinite)
It is called an implicit function as it does not give y explicitly in terms of x unlike
eg 122 yx
122 yx
![Page 7: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/7.jpg)
Implicit Differentiation
Look at the graph of
Even though this is not a function the gradient isclearly defined at every point (except at the twopoints on the x-axis where the gradient is infinite)
It is called an implicit function as it does not give y explicitly in terms of x unlike
eg 122 yx
122 yx
![Page 8: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/8.jpg)
Implicit Differentiation
Look at the graph of
Even though this is not a function the gradient isclearly defined at every point (except at the twopoints on the x-axis where the gradient is infinite)
It is called an implicit function as it does not give y explicitly in terms of x unlike
eg 12 xy
122 yx
![Page 9: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/9.jpg)
As the gradient is clearly defined at every point (except at the two mentioned) we would expect to be able to get an expression for
even though the equation of the circle is not explicit.
dxdy
![Page 10: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/10.jpg)
Example
Find the gradient at the point (2,3) on the circle
1322 yx
Differentiating
022 dxdy
yx
Differentiate x2
![Page 11: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/11.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
022 dxdy
yx
Differentiating
Differentiate y2
y is itself a function of x
So y2 = (y)2 has to be differentiated by the chain rule to give: by differentiating the outer square function and
by differentiating the inner y function.
y2
y
![Page 12: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/12.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
022 dxdy
yx
Differentiating
Differentiate y2
y is itself a function of x
So y2 = (y)2 has to be differentiated by the chain rule to give: by differentiating the outer square function and
by differentiating the inner y function.
y2
y
![Page 13: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/13.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
022 dxdy
yx
Differentiating
Differentiate y2
y is itself a function of x
So y2 = (y)2 has to be differentiated by the chain rule to give: by differentiating the outer square function and
by differentiating the inner y function.
y2
y
![Page 14: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/14.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
022 dxdy
yx
Differentiating
Differentiate y2
y is itself a function of x
So y2 = (y)2 has to be differentiated by the chain rule to give: by differentiating the outer square function and
by differentiating the inner y function.
y2
y
![Page 15: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/15.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
022 dxdy
yx
Differentiating
Differentiate y2
y is itself a function of x
So y2 = (y)2 has to be differentiated by the chain rule to give: by differentiating the outer square function and
by differentiating the inner y function.
y2
y
![Page 16: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/16.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
022 dxdy
yx
Differentiating
Differentiate right hand side
![Page 17: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/17.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
yx
dxdy
xdxdy
y
dxdy
yx
22
022Make the subject
dxdy
![Page 18: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/18.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
yx
dxdy
xdxdy
y
dxdy
yx
22
022Make the subject
dxdy
![Page 19: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/19.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
yx
dxdy
xdxdy
y
dxdy
yx
22
022Make the subject
dxdy
![Page 20: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/20.jpg)
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
yx
dxdy
xdxdy
y
dxdy
yx
22
022
So at (2,3) the gradient is -2/3
![Page 21: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/21.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
This gives a simple rule:
![Page 22: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/22.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
![Page 23: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/23.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
![Page 24: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/24.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
![Page 25: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/25.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
![Page 26: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/26.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
![Page 27: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/27.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
![Page 28: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/28.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
![Page 29: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/29.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
![Page 30: IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a function the gradient is clearly defined at every point](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2c5503460f94a02b3a/html5/thumbnails/30.jpg)
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy