Objectives: 1.Be able to find the derivative using the Constant Rule. 2.Be able to find the...

Objectives:1. Be able to find the derivative using the Constant Rule.

2. Be able to find the derivative using the Power Rule.3. Be able to find the derivative using the Constant Multiple Rule.

4. Be able to find the derivative using the Sum and Difference Rules.

Critical Vocabulary:Constant Rule, Power Rule, Constant Multiple Rule,

Sum and Difference Rules.

I. The Constant Rule

Example: Find the derivative of f(x) = 3 using the definition

x

xfxxfx

)()(lim

0f(x) = 3 f(x + Δx) = 3

xx

33lim

0 xx

0lim

00lim0x

0)(' xf

The derivative of a constant function is zero.

if c is a real number 0][ cdx

d

Example 1: Find the derivative of f(x) = 6 0)(' xf

Example 2: Find the derivative of f(x) = -8 0)(' xf

II. The Power Rule

Example: Find the derivative of each function using the definition

x

xfxxfx

)()(lim

0

1. f(x) = 3x2

2. f(x) = 4x2

3. f(x) = 5x2

1. f’(x) = 6x

2. f’(x) = 8x

3. f’(x) = 10x

1. f(x) = 3x3

2. f(x) = 4x3

3. f(x) = 5x3

1. f’(x) = 9x2

2. f’(x) = 12x2

3. f’(x) = 15x2

What kind of patterns do you observe?

II. The Power Rule

If n is a rational number, then the function f(x) = xn is differentiable and

1 nn nxxdx

d

For f to be differentiable at x = 0, n must be a number such that xn-1 is defined on an interval containing zero

Example 1: Find the derivative of f(x) = x 11' )11()( xxf

1)(' xf

Example 2: Find the derivative of f(x) = x6 16' )61()( xxf

5' 6)( xxf

II. The Power Rule

If n is a rational number, then the function f(x) = xn is differentiable and

1 nn nxxdx

d

For f to be differentiable at x = 0, n must be a number such that xn-1 is defined on an interval containing zero

Sometimes, you need to rewrite an expression if it is not in the form xn (Your final answer may not contain negative exponents)

Example 3: Find the derivative of 3

1)(x

xf 3)( xxf

13)31()(' xxf43)(' xxf

4' 3)(

xxf

II. The Power Rule

If n is a rational number, then the function f(x) = xn is differentiable and

1 nn nxxdx

d

For f to be differentiable at x = 0, n must be a number such that xn-1 is defined on an interval containing zero

Sometimes, you need to rewrite an expression if it is not in the form xn (Your final answer may not contain negative exponents)

Example 4: Find the derivative of xxf )( 2

1

)( xxf 1

2

1

2

11)('

xxf

2

1

2

1)('

xxf

xxf

2

1)('

x

xxf

2)('

II. The Power Rule

If n is a rational number, then the function f(x) = xn is differentiable and

1 nn nxxdx

d

For f to be differentiable at x = 0, n must be a number such that xn-1 is defined on an interval containing zero

Sometimes, you need to rewrite an expression if it is not in the form xn (Your final answer may not contain negative exponents)

Example 5: Use the function f(x) = x2 to find the slope of the tangent line at the point (2, 4).xxf 2)(' General

RuleSlope: 4

III. The Constant Multiple Rule

If f is a differentiable function and c is a real number, then c•f is differentiable and

)(')( xfcxfcdx

d

1)()(' nxcnxf

Example 1: Find the derivative of x

xf2

)( 12)( xxf

11)12()(' xxf

22)(' xxf

2

2)('

xxf

Example 2: Find the derivative of 5

4)(

2xxf 2

5

4)( xxf

1225

4)('

xxfxxf

5

8)('

III. The Constant Multiple Rule

If f is a differentiable function and c is a real number, then c•f is differentiable and

)(')( xfcxfcdx

d

1)()(' nxcnxf

Example 3: Find the derivative of 3 22

1)(

xxf

3

2

2

1)(

xxf

13

2

3

2

2

1)('

xxf

3

5

3

1)('

xxf

3 53

1)('

xxf

3 23

1)('

xxxf

2

3

3)('

x

xxf

IV. The Sum and Difference Rules

)(')(')]()([ xgxfxgxfdx

d

Example 1: Find the slope of the tangent line at (1, -1) of f(x) = x3 - 4x + 2

The sum (or difference) of two differentiable functions is differentiable and is the sum (or difference) of their derivatives.

)(')(')]()([ xgxfxgxfdx

d

f’(x) = 3x2 - 4

m = -1

IV. The Sum and Difference Rules

Example 2: Find the equation of the tangent line to: f(x) = -½x4 + 3x3 – 2x at (-1, -3/2)

f’(x) = -2x3 + 9x2 – 2m = -2(-1)3 + 9(-1)2 – 2m = 2 + 9 – 2m = 9

f(x) = mx + b-3/2 = 9(-1) + b

-3/2 = -9 + b

15/2 = b

f(x) = 9x + 15/2

Part 1: Page 272-273 #3-47 odds

V. Additional Examples

Example 1: Find all the points at which the graph of f(x) = x3 – 3x has horizontal tangent lines.

f’(x) = 3x2 - 3

3x2 - 3 = 0

3x2 = 3

x2 = 1

x = 1 and x = -1

(1,-2) and (-1, 2)

V. Additional Examples

Example 2: Find all the points at which the graph of f(x) = x4 – 4x + 5 has horizontal tangent lines.

f’(x) = 4x3 - 4

4x3 - 4 = 0

4x3 = 4

x3 = 1

x = 1(1, 2)

V. Additional Examples

Example 3: Find k such that the line is tangent to the graph of the function.Function: f(x) = k – x2

Tangent: f(x) = -4x + 7

Equate Functions: k – x2 = -4x + 7Equate Derivatives: -2x = -4

x = 2k – (2)2 = -4(2) + 7

k – 4 = -1

k = 3

Part 1: Page 272-273 #3-47 oddsPart 2: Page 272-273 #49 – 56 all Worksheet 4.2A