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IMPLICIT DIFFERENTIATION
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
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
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
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
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
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
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
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
Example
Find the gradient at the point (2,3) on the circle
1322 yx
Differentiating
022 dxdy
yx
Differentiate x2
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
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
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
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
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
Example contd
Find the gradient at the point (2,3) on the circle
1322 yx
022 dxdy
yx
Differentiating
Differentiate right hand side
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
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
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
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
• Differentiate y terms as though they were x
• Multiply by dxdy
This gives a simple rule:
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
• Differentiate y terms as though they were x
• Multiply by dxdy
22
3
10622
y
xdxdy
dxdy
yx
122 32 yxxExample: Find for
dxdy
