IMPLICIT DIFFERENTIATION. Implicit Differentiation Look at the graph of Even though this is not a...

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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