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2008 W -C L Calculus(I)

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Calculus (I)

WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

2008

(2)

14.3 Partial Derivative

1.

Definition

The partial derivative of f w.r.t. x is defined by

∂f

∂x(x , y ) = lim

h→0

f (x + h, y ) − f (x , y ) h

∂f

∂y(x , y ) = lim

h→0

f (x , y + h) − f (x , y ) h

WEN-CHINGLIEN Calculus (I)

(3)

14.3 Partial Derivative

1.

Definition

The partial derivative of f w.r.t. x is defined by

∂f

∂x(x , y ) = lim

h→0

f (x + h, y ) − f (x , y ) h

∂f

∂y(x , y ) = lim

h→0

f (x , y + h) − f (x , y ) h

(4)

14.3 Partial Derivative

1.

Definition

The partial derivative of f w.r.t. x is defined by

∂f

∂x(x , y ) = lim

h→0

f (x + h, y ) − f (x , y ) h

∂f

∂y(x , y ) = lim

h→0

f (x , y + h) − f (x , y ) h

WEN-CHINGLIEN Calculus (I)

(5)

14.3 Partial Derivative

1.

Definition

The partial derivative of f w.r.t. x is defined by

∂f

∂x(x , y ) = lim

h→0

f (x + h, y ) − f (x , y ) h

∂f

∂y(x , y ) = lim

h→0

f (x , y + h) − f (x , y ) h

(6)

Example 1:

(1) Find ∂f

∂x, ∂f

∂y for f (x , y ) = exy (2) f (x , y ) = xy

x2+2, ∂f

∂x(1, 2) =?, ∂f

∂y(1, 2) =?

(3) f (x , y ) = sin(x2+y )

WEN-CHINGLIEN Calculus (I)

(7)

Example 1:

(1) Find ∂f

∂x, ∂f

∂y for f (x , y ) = exy (2) f (x , y ) = xy

x2+2, ∂f

∂x(1, 2) =?, ∂f

∂y(1, 2) =?

(3) f (x , y ) = sin(x2+y )

(8)

Example 1:

(1) Find ∂f

∂x, ∂f

∂y for f (x , y ) = exy (2) f (x , y ) = xy

x2+2, ∂f

∂x(1, 2) =?, ∂f

∂y(1, 2) =?

(3) f (x , y ) = sin(x2+y )

WEN-CHINGLIEN Calculus (I)

(9)

Example 1:

(1) Find ∂f

∂x, ∂f

∂y for f (x , y ) = exy (2) f (x , y ) = xy

x2+2, ∂f

∂x(1, 2) =?, ∂f

∂y(1, 2) =?

(3) f (x , y ) = sin(x2+y )

(10)

2.

Geometric Interpretation

WEN-CHINGLIEN Calculus (I)

(11)

2.

Geometric Interpretation

(12)

3.

Introduction of higher order partial derivatives

2

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



2

f

∂ x

2

2

f

∂ y

2

...

WEN-CHINGLIEN Calculus (I)

(13)

3.

Introduction of higher order partial derivatives

2

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



2

f

∂ x

2

2

f

∂ y

2

...

(14)

3.

Introduction of higher order partial derivatives

2

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



2

f

∂ x

2

2

f

∂ y

2

...

WEN-CHINGLIEN Calculus (I)

(15)

3.

Introduction of higher order partial derivatives

2

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



2

f

∂ x

2

2

f

∂ y

2

...

(16)

3.

Introduction of higher order partial derivatives

2

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



2

f

∂ x

2

2

f

∂ y

2

...

WEN-CHINGLIEN Calculus (I)

(17)

3.

Introduction of higher order partial derivatives

2

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



2

f

∂ x

2

2

f

∂ y

2

...

(18)

Theorem (The mixed derivative theorem) If f (x , y ) and its partial derivatives fx, fy, fxy, fyx

are continuous on an open disk centered at the point (x0,y0),then

fxy(x0,y0) =fyx(x0,y0)

WEN-CHINGLIEN Calculus (I)

(19)

Theorem (The mixed derivative theorem) If f (x , y ) and its partial derivatives fx, fy, fxy, fyx

are continuous on an open disk centered at the point (x0,y0),then

fxy(x0,y0) =fyx(x0,y0)

(20)

Theorem (The mixed derivative theorem) If f (x , y ) and its partial derivatives fx, fy, fxy, fyx

are continuous on an open disk centered at the point (x0,y0),then

fxy(x0,y0) =fyx(x0,y0)

WEN-CHINGLIEN Calculus (I)

(21)

Theorem (The mixed derivative theorem) If f (x , y ) and its partial derivatives fx, fy, fxy, fyx

are continuous on an open disk centered at the point (x0,y0),then

fxy(x0,y0) =fyx(x0,y0)

(22)

Thank you.

WEN-CHINGLIEN Calculus (I)

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