2008 W -C L Calculus(I)

Calculus (I)

W^EN-C^HINGL^IEN

Department of Mathematics National Cheng Kung University

2008

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)

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

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)

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

Example 1:

(1) Find ∂f

∂x, ∂f

∂y for f (x , y ) = e^xy (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)

Example 1:

(1) Find ∂f

∂x, ∂f

∂y for f (x , y ) = e^xy (2) f (x , y ) = xy

x²+2, ∂f

∂x(1, 2) =?, ∂f

∂y(1, 2) =?

(3) f (x , y ) = sin(x²+y )

Example 1:

(1) Find ∂f

∂x, ∂f

∂y for f (x , y ) = e^xy (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)

Example 1:

(1) Find ∂f

∂x, ∂f

∂y for f (x , y ) = e^xy (2) f (x , y ) = xy

x²+2, ∂f

∂x(1, 2) =?, ∂f

∂y(1, 2) =?

(3) f (x , y ) = sin(x²+y )

2.

Geometric Interpretation

WEN-CHINGLIEN Calculus (I)

2.

Geometric Interpretation

3.

Introduction of higher order partial derivatives

∂

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



∂

f

∂ x

∂

f

∂ y

...

WEN-CHINGLIEN Calculus (I)

3.

Introduction of higher order partial derivatives

∂

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



∂

f

∂ x

∂

f

∂ y

...

3.

Introduction of higher order partial derivatives

∂

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



∂

f

∂ x

∂

f

∂ y

...

WEN-CHINGLIEN Calculus (I)

3.

Introduction of higher order partial derivatives

∂

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



∂

f

∂ x

∂

f

∂ y

...

3.

Introduction of higher order partial derivatives

∂

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



∂

f

∂ x

∂

f

∂ y

...

WEN-CHINGLIEN Calculus (I)

3.

Introduction of higher order partial derivatives

∂

f

∂ x ∂y = ∂

∂x

 ∂f

∂y



∂

f

∂ x

∂

f

∂ y

...

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)

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)

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)

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)

Thank you.

WEN-CHINGLIEN Calculus (I)