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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
