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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

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## 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)

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## 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

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## 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

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### 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)

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### 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 )

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### Geometric Interpretation

WEN-CHINGLIEN Calculus (I)

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(12)

2

2

2

2

2

### ...

WEN-CHINGLIEN Calculus (I)

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2

2

2

2

2

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2

2

2

2

2

### ...

WEN-CHINGLIEN Calculus (I)

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2

2

2

2

2

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2

2

2

2

2

### ...

WEN-CHINGLIEN Calculus (I)

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2

2

2

2

2

### ...

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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)

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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)

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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)

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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)

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## Thank you.

WEN-CHINGLIEN Calculus (I)

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

W EN -C HING L IEN Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

W EN -C HING L IEN Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

W EN -C HING L IEN Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

W EN -C HING L IEN Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung