calc 2.2a
TRANSCRIPT
![Page 1: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/1.jpg)
2.2 BASIC DIFFERENTIATION RULESTo Find:•Derivative using Constant Rule•Derivative using Power Rule•Derivative using Constant Multiple Rule•Derivative using Sum and Difference Rules•Derivative for Sine and Cosine
![Page 2: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/2.jpg)
Makes sense, right?
![Page 3: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/3.jpg)
Let’s see if we can come up with the power rule.
a. f(x) = x
b. f(x) = x2
c. f(x) = x3
Do you recognize a pattern?
2 2
0
( ) ( ) ( )'( ) lim
x
f x x f x x x xf x
x x
0
( ) ( ) ( )'( ) lim lim 1
x
f x x f x x x xf x
x x
22 2
0
2 (2 )lim lim 2
x
x x x x x x x xx
x x
2 3 23 2 3 22
0
3 3 (3 3 )lim lim 3
x
x x x x x x x x x x x xx
x x
3 3
0
( ) ( ) ( )'( ) lim lim
x
f x x f x x x xf x
x x
![Page 4: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/4.jpg)
a. f(x) = x5
b. c.
Sometimes you need to rewrite to different form!
5( )g x x
4
1y
x
Find the following derivatives:
![Page 5: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/5.jpg)
Ex. 3 p. 109 Find the slope of a graph.
Find the slope of f(x) = x4 when a.x = -1b.x = 0c.x = 1
3'( ) 4f x x
3'( 1) 4 1 4f
3'(0) 4 0 0f
3'(1) 4 1 4f
![Page 6: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/6.jpg)
Ex 4 p. 109 Finding an Equation of a Tangent Line
Find the equation of the tangent line to the graph of f(x) = x3 when x = -2
To find an equation of a line, we need a point and a slope. The point we are looking at is (-2, f(-2)). In other words, find the y-value in the original function!f(-2) = (-2)3 = -8. So our point of tangency is (-2, -8)
Next we need a slope. Find the derivative & evaluate.f ‘(x) = 3x2 so find f ‘(-2) = 3٠(-2)2=12
Equation: (y – (-8)) = 12(x –(-2))So y = 12x +16 is the equation of the tangent line.
![Page 7: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/7.jpg)
Informally, this states that constants can be factored out of the differentiation process.
2y
x 1 2
2
22 2
dy dx x
dx dx x
24( )
5
tf t
24 4 8'( ) 2
5 5 5
df t t t t
dt
Ex 5 p. 110 Using the Constant Multiple Rule
![Page 8: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/8.jpg)
Ex 5 continued
2y x1 1
2 21 1
' 2 22
dy x x
dx x
3 2
1
2y
x
523 3
5 3 53
1 1 2 1 1'
2 2 3 33
dy x x or
dx xx
3
2
xy 3 3 3
' 1 2 2 2
dy x
dx
![Page 9: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/9.jpg)
Ex 6 p. 111 Using Parentheses when differentiating
Original function Rewrite Differentiate Simplify
2
2
5y
x 22
5y x 32
' 25
y x 3
4'
5y
x
2
2
5y
x 22
25y x 32
' 225
y x 3
4'
25y
x
3
5
4y
x 35
4y x 25
' 34
y x215
'4
xy
3
5
4y
x 3320y x 2' 320 3y x 2' 960y x
![Page 10: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/10.jpg)
This can be expanded to any number of functionsEx 7, p. 111 Using Sum and Difference Rules
a. 3( ) 2 5f x x x 2'( ) 3 2f x x 4
3( ) 4 32
xg x x x b.
3 2'( ) 2 12 3g x x x
![Page 11: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/11.jpg)
Proof of derivative of sine:
0
sin( ) sin sin cos cos sin sinsin lim lim
x
d x x x x x x x xx
dx x x
cos sin sin 1 coslim
x x x x
x
0
1 cossinlim cos sinx
xxx x
x x
cos (1) sin (0) cosx x x
![Page 12: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/12.jpg)
![Page 13: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/13.jpg)
Last but not least, Ex 8, p112, Derivatives of sine and cosine
Function Derivative
5siny x ' 5cosy x
sin 1sin
5 5
xy x
1 cos' cos
5 5
xy x
3cosy x x ' 1 3siny x
![Page 14: Calc 2.2a](https://reader035.vdocuments.site/reader035/viewer/2022062514/557c279dd8b42a925b8b5047/html5/thumbnails/14.jpg)
Assign: 2.2a p. 115 #1-65 every other odd
Heads up – each of you will need to create a derivative project – something that you will use to remember all the derivative rules we learn in this chapter. This will be due Monday Oct 17. See paper for details. (online too)