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WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

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## 4.1 The Derivative

Definition

A real function f is said to be differentiable at a point a ∈R if and only if f is defined on some open interval I

containing a and

(1) f0(a) = lim

h→0

f (a + h) − f (a) h

exists. In this case f0(a) is called the derivative of f at a.

Dxf = df

dx =f(1)=f0

(3)

## 4.1 The Derivative

Definition

A real function f is said to be differentiable at a point a ∈R if and only if f is defined on some open interval I

containing a and

(1) f0(a) = lim

h→0

f (a + h) − f (a) h

exists. In this case f0(a) is called the derivative of f at a.

Dxf = df

dx =f(1)=f0

(4)

## 4.1 The Derivative

Definition

A real function f is said to be differentiable at a point a ∈R if and only if f is defined on some open interval I

containing a and

(1) f0(a) = lim

h→0

f (a + h) − f (a) h

exists. In this case f0(a) is called the derivative of f at a.

### Notation:

Dxf = df

dx =f(1)=f0

(5)

Theorem

A real function f is differentiable at some point a ∈R if and only if there exists an open interval I and a function

F : I →R such that a ∈ I, f is defined on I, F is continuous at a, and

(3) f (x ) = F (x )(x − a) + f (a) holds for all x ∈ I, in which case F (a) = f0(a).

(6)

Theorem

A real function f is differentiable at some point a ∈R if and only if there exists an open interval I and a function

F : I →R such that a ∈ I, f is defined on I, F is continuous at a, and

(3) f (x ) = F (x )(x − a) + f (a) holds for all x ∈ I, in which case F (a) = f0(a).

(7)

Theorem

Let f :R → R. Then f is differentiable at a if and only if there is a function T of the form T (x ) := mx such that

(4) lim

h→0

|f (a + h) − f (a) − T (h)|

|h| =0.

(8)

Theorem

Let f :R → R. Then f is differentiable at a if and only if there is a function T of the form T (x ) := mx such that

(4) lim

h→0

|f (a + h) − f (a) − T (h)|

|h| =0.

(9)

Theorem

If f is differentiable at a, then f is continuous at a.

(10)

Theorem

If f is differentiable at a, then f is continuous at a.

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Definition

Let I be a nondegenerate interval.

(i)

A function f : I →R is said to be differentiable on I if and only if

f0I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.

(12)

Definition

Let I be a nondegenerate interval.

(i)

A function f : I →R is said to be differentiable on I if and only if

f0I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.

(13)

Definition

Let I be a nondegenerate interval.

(i)

A function f : I →R is said to be differentiable on I if and only if

f0I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.

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Definition

Let I be a nondegenerate interval.

(i)

A function f : I →R is said to be differentiable on I if and only if

f0I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.

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Definition

Let I be a nondegenerate interval.

(i)

A function f : I →R is said to be differentiable on I if and only if

f0I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.

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Definition

Let I be a nondegenerate interval.

(i)

A function f : I →R is said to be differentiable on I if and only if

f0I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if f0I exists and is continuous on I.

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### Example:

The function

f (x ) = (

x2sin(1

x) x 6= 0

0 x = 0

is differentiable onR but not continuously differentiable on any interival that contains the origin.

(18)

### Example:

The function

f (x ) = (

x2sin(1

x) x 6= 0

0 x = 0

is differentiable onR but not continuously differentiable on any interival that contains the origin.

(19)

### Remark:

f (x ) = |x | is differentiable on [0, 1] and on [−1, 0] but not on [−1, 1].

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### Remark:

f (x ) = |x | is differentiable on [0, 1] and on [−1, 0] but not on [−1, 1].

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0,it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5,f is not differentiable at x = 0. However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0.However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5,f is not differentiable at x = 0. However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0.However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore,f is differentiable on [0, 1] and on [−1, 0]. 2

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore,f is differentiable on [0, 1] and on [−1, 0]. 2

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### Proof:

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f0(x ) = 1 for x > 0 and f0(x ) = −1 for x < 0). By example 4.5, f is not differentiable at x = 0. However,

f0[0,1](0) = lim

h→0+

|h|

h =1 and f0[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

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## Thank you.

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