3.3 Differentiation Rules -...
Transcript of 3.3 Differentiation Rules -...
![Page 1: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/1.jpg)
§ 3.3 Differentiation Rules
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Constant Functions
What does a constant function look like?
x
y
y = 2
What is the slope?
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Constant Functions
What does a constant function look like?
x
y
y = 2
What is the slope?
![Page 4: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/4.jpg)
Constant Functions
What does a constant function look like?
x
y
y = 2
What is the slope?
![Page 5: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/5.jpg)
Constant Functions
What does a constant function look like?
x
y
y = 2
y = 0
Constant Function Ruleddx c = 0 for all c ∈ R.
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Constant Function Rule
Why?
f ′(x) = limh→0
f (x + h)− f (x)
h
= limh→0
c− ch= 0
![Page 7: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/7.jpg)
Constant Function Rule
Why?
f ′(x) = limh→0
f (x + h)− f (x)
h
= limh→0
c− ch= 0
![Page 8: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/8.jpg)
Constant Function Rule
Why?
f ′(x) = limh→0
f (x + h)− f (x)
h
= limh→0
c− ch
= 0
![Page 9: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/9.jpg)
Constant Function Rule
Why?
f ′(x) = limh→0
f (x + h)− f (x)
h
= limh→0
c− ch= 0
![Page 10: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/10.jpg)
Linear Functions
What is the slope of a linear function?
x
y
y = 12 x− 1
What is the slope?
![Page 11: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/11.jpg)
Linear Functions
What is the slope of a linear function?
x
y
y = 12 x− 1
What is the slope?
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Linear Functions
What is the slope of a linear function?
x
y
y = 12 x− 1
What is the slope?
![Page 13: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/13.jpg)
Linear Functions
What is the slope of a linear function?
x
y
y = 12 x− 1
y = 12
![Page 14: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/14.jpg)
Quadratic Functions
What does a quadratic function look like?
x
y
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Quadratic Functions
What does a quadratic function look like?
x
y
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Quadratic Functions
What is the slope at a point?
x
y
y = x2 − 1
y = 2x
![Page 17: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/17.jpg)
Quadratic Functions
What is the slope at a point?
x
y
y = x2 − 1
y = 2x
![Page 18: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/18.jpg)
Quadratic Functions
What is the slope at a point?
x
y
y = x2 − 1
y = 2x
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Cubic Functions?
x
y
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Cubic Functions?
x
y
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Power Functions
ddx
xn = limh→0
(x + h)n − xn
h
= limh→0
xn + nxn−1h + n(n−1)2 xn−2h2 + . . . + hn − xn
h
= limh→0
nxn−1h + n(n−1)2 xn−2h2 + . . . + hn
h
= limh→0
h(nxn−1 + n(n−1)2 xn−2h + . . . + hn−1)
h
= limh→0
nxn−1 +n(n− 1)
2xn−2h + . . . + hn−1
= nxn−1
![Page 22: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/22.jpg)
Power Functions
ddx
xn = limh→0
(x + h)n − xn
h
= limh→0
xn + nxn−1h + n(n−1)2 xn−2h2 + . . . + hn − xn
h
= limh→0
nxn−1h + n(n−1)2 xn−2h2 + . . . + hn
h
= limh→0
h(nxn−1 + n(n−1)2 xn−2h + . . . + hn−1)
h
= limh→0
nxn−1 +n(n− 1)
2xn−2h + . . . + hn−1
= nxn−1
![Page 23: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/23.jpg)
Power Functions
ddx
xn = limh→0
(x + h)n − xn
h
= limh→0
xn + nxn−1h + n(n−1)2 xn−2h2 + . . . + hn − xn
h
= limh→0
nxn−1h + n(n−1)2 xn−2h2 + . . . + hn
h
= limh→0
h(nxn−1 + n(n−1)2 xn−2h + . . . + hn−1)
h
= limh→0
nxn−1 +n(n− 1)
2xn−2h + . . . + hn−1
= nxn−1
![Page 24: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/24.jpg)
Power Functions
ddx
xn = limh→0
(x + h)n − xn
h
= limh→0
xn + nxn−1h + n(n−1)2 xn−2h2 + . . . + hn − xn
h
= limh→0
nxn−1h + n(n−1)2 xn−2h2 + . . . + hn
h
= limh→0
h(nxn−1 + n(n−1)2 xn−2h + . . . + hn−1)
h
= limh→0
nxn−1 +n(n− 1)
2xn−2h + . . . + hn−1
= nxn−1
![Page 25: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/25.jpg)
Power Functions
ddx
xn = limh→0
(x + h)n − xn
h
= limh→0
xn + nxn−1h + n(n−1)2 xn−2h2 + . . . + hn − xn
h
= limh→0
nxn−1h + n(n−1)2 xn−2h2 + . . . + hn
h
= limh→0
h(nxn−1 + n(n−1)2 xn−2h + . . . + hn−1)
h
= limh→0
nxn−1 +n(n− 1)
2xn−2h + . . . + hn−1
= nxn−1
![Page 26: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/26.jpg)
Power Functions
ddx
xn = limh→0
(x + h)n − xn
h
= limh→0
xn + nxn−1h + n(n−1)2 xn−2h2 + . . . + hn − xn
h
= limh→0
nxn−1h + n(n−1)2 xn−2h2 + . . . + hn
h
= limh→0
h(nxn−1 + n(n−1)2 xn−2h + . . . + hn−1)
h
= limh→0
nxn−1 +n(n− 1)
2xn−2h + . . . + hn−1
= nxn−1
![Page 27: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/27.jpg)
Power Functions
Derivative of Power Functionsddx
xn = nxn−1
for all n ∈ R.
How does this work with negative numbers? Fractions?
![Page 28: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/28.jpg)
Power Functions
Derivative of Power Functionsddx
xn = nxn−1
for all n ∈ R.
How does this work with negative numbers? Fractions?
![Page 29: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/29.jpg)
Examples
Example
Find ddx
1x .
ddx
1x
=ddx
x−1
= −x−2
=−1x2
![Page 30: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/30.jpg)
Examples
Example
Find ddx
1x .
ddx
1x
=
ddx
x−1
= −x−2
=−1x2
![Page 31: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/31.jpg)
Examples
Example
Find ddx
1x .
ddx
1x
=ddx
x−1
= −x−2
=−1x2
![Page 32: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/32.jpg)
Examples
Example
Find ddx
1x .
ddx
1x
=ddx
x−1
= −x−2
=−1x2
![Page 33: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/33.jpg)
Examples
Example
Find ddx
1x .
ddx
1x
=ddx
x−1
= −x−2
=−1x2
![Page 34: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/34.jpg)
Examples
Example
Find ddx√
x.
ddx√
x =ddx
x12
=12
x( 12−1)
=1
2x12
=1
2√
x
![Page 35: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/35.jpg)
Examples
Example
Find ddx√
x.
ddx√
x =
ddx
x12
=12
x( 12−1)
=1
2x12
=1
2√
x
![Page 36: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/36.jpg)
Examples
Example
Find ddx√
x.
ddx√
x =ddx
x12
=12
x( 12−1)
=1
2x12
=1
2√
x
![Page 37: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/37.jpg)
Examples
Example
Find ddx√
x.
ddx√
x =ddx
x12
=12
x( 12−1)
=1
2x12
=1
2√
x
![Page 38: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/38.jpg)
Examples
Example
Find ddx√
x.
ddx√
x =ddx
x12
=12
x( 12−1)
=1
2x12
=1
2√
x
![Page 39: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/39.jpg)
Examples
Example
Find ddx√
x.
ddx√
x =ddx
x12
=12
x( 12−1)
=1
2x12
=1
2√
x
![Page 40: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/40.jpg)
Constant Multiple Rule
Example
Find ddx 3x2.
ddx
3x2 = limh→0
3(x + h)2 − 3x2
h
= limh→0
3((x + h)2 − x2)
h
= limh→0
3(x2 + 2xh + h2 − x2)
h
= limh→0
3h(2xh + h2)
h= lim
h→03(2x + h)
= 3(2x) = 6
![Page 41: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/41.jpg)
Constant Multiple Rule
Example
Find ddx 3x2.
ddx
3x2 = limh→0
3(x + h)2 − 3x2
h
= limh→0
3((x + h)2 − x2)
h
= limh→0
3(x2 + 2xh + h2 − x2)
h
= limh→0
3h(2xh + h2)
h= lim
h→03(2x + h)
= 3(2x) = 6
![Page 42: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/42.jpg)
Constant Multiple Rule
Example
Find ddx 3x2.
ddx
3x2 = limh→0
3(x + h)2 − 3x2
h
= limh→0
3((x + h)2 − x2)
h
= limh→0
3(x2 + 2xh + h2 − x2)
h
= limh→0
3h(2xh + h2)
h= lim
h→03(2x + h)
= 3(2x) = 6
![Page 43: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/43.jpg)
Constant Multiple Rule
Example
Find ddx 3x2.
ddx
3x2 = limh→0
3(x + h)2 − 3x2
h
= limh→0
3((x + h)2 − x2)
h
= limh→0
3(x2 + 2xh + h2 − x2)
h
= limh→0
3h(2xh + h2)
h= lim
h→03(2x + h)
= 3(2x) = 6
![Page 44: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/44.jpg)
Constant Multiple Rule
Example
Find ddx 3x2.
ddx
3x2 = limh→0
3(x + h)2 − 3x2
h
= limh→0
3((x + h)2 − x2)
h
= limh→0
3(x2 + 2xh + h2 − x2)
h
= limh→0
3h(2xh + h2)
h
= limh→0
3(2x + h)
= 3(2x) = 6
![Page 45: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/45.jpg)
Constant Multiple Rule
Example
Find ddx 3x2.
ddx
3x2 = limh→0
3(x + h)2 − 3x2
h
= limh→0
3((x + h)2 − x2)
h
= limh→0
3(x2 + 2xh + h2 − x2)
h
= limh→0
3h(2xh + h2)
h= lim
h→03(2x + h)
= 3(2x) = 6
![Page 46: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/46.jpg)
Constant Multiple Rule
Example
Find ddx 3x2.
ddx
3x2 = limh→0
3(x + h)2 − 3x2
h
= limh→0
3((x + h)2 − x2)
h
= limh→0
3(x2 + 2xh + h2 − x2)
h
= limh→0
3h(2xh + h2)
h= lim
h→03(2x + h)
= 3(2x) = 6
![Page 47: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/47.jpg)
Constant Multiples
Constant Multiple Rule
For any differentiable function y = f (x),
ddx
cf (x) = cddx
f (x) = cf ′(x)
![Page 48: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/48.jpg)
Sums and Differences
Sum and Difference Ruleddx
[f (x)± g(x)] = f ′(x)± g′(x)
limh→0
f (x + h)± g(x + h)− (f (x)± g(x))
h
= limh→0
f (x + h)− f (x)
h± g(x + h)− g(x)
h
= limh→0
f (x + h)− f (x)
h± lim
h→0
g(x + h)− g(x)
h= f ′(x)± g′(x)
![Page 49: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/49.jpg)
Sums and Differences
Sum and Difference Ruleddx
[f (x)± g(x)] = f ′(x)± g′(x)
limh→0
f (x + h)± g(x + h)− (f (x)± g(x))
h
= limh→0
f (x + h)− f (x)
h± g(x + h)− g(x)
h
= limh→0
f (x + h)− f (x)
h± lim
h→0
g(x + h)− g(x)
h= f ′(x)± g′(x)
![Page 50: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/50.jpg)
Sums and Differences
Sum and Difference Ruleddx
[f (x)± g(x)] = f ′(x)± g′(x)
limh→0
f (x + h)± g(x + h)− (f (x)± g(x))
h
= limh→0
f (x + h)− f (x)
h± g(x + h)− g(x)
h
= limh→0
f (x + h)− f (x)
h± lim
h→0
g(x + h)− g(x)
h= f ′(x)± g′(x)
![Page 51: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/51.jpg)
Sums and Differences
Sum and Difference Ruleddx
[f (x)± g(x)] = f ′(x)± g′(x)
limh→0
f (x + h)± g(x + h)− (f (x)± g(x))
h
= limh→0
f (x + h)− f (x)
h± g(x + h)− g(x)
h
= limh→0
f (x + h)− f (x)
h± lim
h→0
g(x + h)− g(x)
h
= f ′(x)± g′(x)
![Page 52: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/52.jpg)
Sums and Differences
Sum and Difference Ruleddx
[f (x)± g(x)] = f ′(x)± g′(x)
limh→0
f (x + h)± g(x + h)− (f (x)± g(x))
h
= limh→0
f (x + h)− f (x)
h± g(x + h)− g(x)
h
= limh→0
f (x + h)− f (x)
h± lim
h→0
g(x + h)− g(x)
h= f ′(x)± g′(x)
![Page 53: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/53.jpg)
An Example
Example
Find f ′(x) for f (x) = −2x2 + 4x3 − 3x .
f ′(x) = − 4x + 12x2 +3x2
![Page 54: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/54.jpg)
An Example
Example
Find f ′(x) for f (x) = −2x2 + 4x3 − 3x .
f ′(x) =
− 4x + 12x2 +3x2
![Page 55: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/55.jpg)
An Example
Example
Find f ′(x) for f (x) = −2x2 + 4x3 − 3x .
f ′(x) = − 4x
+ 12x2 +3x2
![Page 56: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/56.jpg)
An Example
Example
Find f ′(x) for f (x) = −2x2 + 4x3 − 3x .
f ′(x) = − 4x + 12x2
+3x2
![Page 57: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/57.jpg)
An Example
Example
Find f ′(x) for f (x) = −2x2 + 4x3 − 3x .
f ′(x) = − 4x + 12x2 +3x2
![Page 58: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/58.jpg)
Product Rule
Let ∆f = f (x + h)− f (x) and ∆g = g(x + h)− g(x).
f (x)
g(x)
∆f
∆g
![Page 59: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/59.jpg)
Product Rule
Let ∆f = f (x + h)− f (x) and ∆g = g(x + h)− g(x).
f (x)
g(x)
∆f
∆g
![Page 60: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/60.jpg)
Product Rule
Let ∆f = f (x + h)− f (x) and ∆g = g(x + h)− g(x).
f (x)
g(x)
∆f
∆g
![Page 61: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/61.jpg)
Product Rule
Let ∆f = f (x + h)− f (x) and ∆g = g(x + h)− g(x).
f (x)
g(x)
∆f
∆g
![Page 62: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/62.jpg)
Product Rule
Let ∆f = f (x + h)− f (x) and ∆g = g(x + h)− g(x).
f (x)
g(x)
∆f
∆g
![Page 63: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/63.jpg)
Product Rule
Consider f (x + h)g(x + h).
f (x + h)f (x)
g(x + h)
g(x)
f (x)g(x) g(x)∆f
f (x)∆g ∆f ∆g
![Page 64: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/64.jpg)
Product Rule
Consider f (x + h)g(x + h).
f (x + h)f (x)
g(x + h)
g(x)
f (x)g(x) g(x)∆f
f (x)∆g ∆f ∆g
![Page 65: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/65.jpg)
Product Rule
Consider f (x + h)g(x + h).
f (x + h)f (x)
g(x + h)
g(x)
f (x)g(x)
g(x)∆f
f (x)∆g ∆f ∆g
![Page 66: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/66.jpg)
Product Rule
Consider f (x + h)g(x + h).
f (x + h)f (x)
g(x + h)
g(x)
f (x)g(x) g(x)∆f
f (x)∆g ∆f ∆g
![Page 67: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/67.jpg)
Product Rule
Consider f (x + h)g(x + h).
f (x + h)f (x)
g(x + h)
g(x)
f (x)g(x) g(x)∆f
f (x)∆g
∆f ∆g
![Page 68: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/68.jpg)
Product Rule
Consider f (x + h)g(x + h).
f (x + h)f (x)
g(x + h)
g(x)
f (x)g(x) g(x)∆f
f (x)∆g ∆f ∆g
![Page 69: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/69.jpg)
As h→ 0
![Page 70: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/70.jpg)
As h→ 0
![Page 71: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/71.jpg)
As h→ 0
![Page 72: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/72.jpg)
The Product Rule
ddx
[f (x)g(x)] = limh→0
f (x + h)g(x + h)− f (x)g(x)
h
= limh→0
(∆fh
g(x) + f (x)∆gh
+∆f ∆g
h
)= f ′(x)g(x) + g′(x)f (x) + lim
h→0
(∆fh
∆gh
h)
= f ′(x)g(x) + g′(x)f (x)
![Page 73: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/73.jpg)
The Product Rule
ddx
[f (x)g(x)] = limh→0
f (x + h)g(x + h)− f (x)g(x)
h
= limh→0
(∆fh
g(x) + f (x)∆gh
+∆f ∆g
h
)
= f ′(x)g(x) + g′(x)f (x) + limh→0
(∆fh
∆gh
h)
= f ′(x)g(x) + g′(x)f (x)
![Page 74: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/74.jpg)
The Product Rule
ddx
[f (x)g(x)] = limh→0
f (x + h)g(x + h)− f (x)g(x)
h
= limh→0
(∆fh
g(x) + f (x)∆gh
+∆f ∆g
h
)= f ′(x)g(x) + g′(x)f (x) + lim
h→0
(∆fh
∆gh
h)
= f ′(x)g(x) + g′(x)f (x)
![Page 75: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/75.jpg)
The Product Rule
ddx
[f (x)g(x)] = limh→0
f (x + h)g(x + h)− f (x)g(x)
h
= limh→0
(∆fh
g(x) + f (x)∆gh
+∆f ∆g
h
)= f ′(x)g(x) + g′(x)f (x) + lim
h→0
(∆fh
∆gh
h)
= f ′(x)g(x) + g′(x)f (x)
![Page 76: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/76.jpg)
The Product Rule
The Product Ruleddx f (x)g(x) = f (x)g′(x) + g(x)f ′(x)
Alternate StatementIf u = f (x) and v = g(x), then
ddx
uv = uv′ + vu′
![Page 77: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/77.jpg)
The Product Rule
The Product Ruleddx f (x)g(x) = f (x)g′(x) + g(x)f ′(x)
Alternate StatementIf u = f (x) and v = g(x), then
ddx
uv = uv′ + vu′
![Page 78: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/78.jpg)
An Example
Example
Find f ′(x) for f (x) = x2(x− 4)5.
f ′(x) = x2 ddx
(x− 4)5 + (x− 4)5 ddx
x2
= x2(5(x− 4)4) + (x− 4)5(2x)
= x(x− 4)4(5x + (x− 4)(2))
= x(x− 4)4(5x + 2x− 8)
= x(x− 4)4(7x− 8)
![Page 79: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/79.jpg)
An Example
Example
Find f ′(x) for f (x) = x2(x− 4)5.
f ′(x) = x2 ddx
(x− 4)5 + (x− 4)5 ddx
x2
= x2(5(x− 4)4) + (x− 4)5(2x)
= x(x− 4)4(5x + (x− 4)(2))
= x(x− 4)4(5x + 2x− 8)
= x(x− 4)4(7x− 8)
![Page 80: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/80.jpg)
An Example
Example
Find f ′(x) for f (x) = x2(x− 4)5.
f ′(x) = x2 ddx
(x− 4)5 + (x− 4)5 ddx
x2
= x2(5(x− 4)4) + (x− 4)5(2x)
= x(x− 4)4(5x + (x− 4)(2))
= x(x− 4)4(5x + 2x− 8)
= x(x− 4)4(7x− 8)
![Page 81: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/81.jpg)
An Example
Example
Find f ′(x) for f (x) = x2(x− 4)5.
f ′(x) = x2 ddx
(x− 4)5 + (x− 4)5 ddx
x2
= x2(5(x− 4)4) + (x− 4)5(2x)
= x(x− 4)4(5x + (x− 4)(2))
= x(x− 4)4(5x + 2x− 8)
= x(x− 4)4(7x− 8)
![Page 82: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/82.jpg)
An Example
Example
Find f ′(x) for f (x) = x2(x− 4)5.
f ′(x) = x2 ddx
(x− 4)5 + (x− 4)5 ddx
x2
= x2(5(x− 4)4) + (x− 4)5(2x)
= x(x− 4)4(5x + (x− 4)(2))
= x(x− 4)4(5x + 2x− 8)
= x(x− 4)4(7x− 8)
![Page 83: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/83.jpg)
An Example
Example
Find f ′(x) for f (x) = x2(x− 4)5.
f ′(x) = x2 ddx
(x− 4)5 + (x− 4)5 ddx
x2
= x2(5(x− 4)4) + (x− 4)5(2x)
= x(x− 4)4(5x + (x− 4)(2))
= x(x− 4)4(5x + 2x− 8)
= x(x− 4)4(7x− 8)
![Page 84: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/84.jpg)
The Quotient Rule
Let Q(x) = f (x)g(x) .
f (x) = Q(x)g(x)
f ′(x) = Q(x)g′(x) + Q′(x)g(x)
f ′(x) =f (x)g′(x)
g(x)+ Q′(x)g(x)
f ′(x)g(x) = f (x)g′(x) + Q′(x)[g(x)]2
Q′(x) =f ′(x)g(x)− g′(x)f (x)
[g(x)]2
![Page 85: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/85.jpg)
The Quotient Rule
Let Q(x) = f (x)g(x) .
f (x) = Q(x)g(x)
f ′(x) = Q(x)g′(x) + Q′(x)g(x)
f ′(x) =f (x)g′(x)
g(x)+ Q′(x)g(x)
f ′(x)g(x) = f (x)g′(x) + Q′(x)[g(x)]2
Q′(x) =f ′(x)g(x)− g′(x)f (x)
[g(x)]2
![Page 86: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/86.jpg)
The Quotient Rule
Let Q(x) = f (x)g(x) .
f (x) = Q(x)g(x)
f ′(x) = Q(x)g′(x) + Q′(x)g(x)
f ′(x) =f (x)g′(x)
g(x)+ Q′(x)g(x)
f ′(x)g(x) = f (x)g′(x) + Q′(x)[g(x)]2
Q′(x) =f ′(x)g(x)− g′(x)f (x)
[g(x)]2
![Page 87: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/87.jpg)
The Quotient Rule
Let Q(x) = f (x)g(x) .
f (x) = Q(x)g(x)
f ′(x) = Q(x)g′(x) + Q′(x)g(x)
f ′(x) =f (x)g′(x)
g(x)+ Q′(x)g(x)
f ′(x)g(x) = f (x)g′(x) + Q′(x)[g(x)]2
Q′(x) =f ′(x)g(x)− g′(x)f (x)
[g(x)]2
![Page 88: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/88.jpg)
The Quotient Rule
Let Q(x) = f (x)g(x) .
f (x) = Q(x)g(x)
f ′(x) = Q(x)g′(x) + Q′(x)g(x)
f ′(x) =f (x)g′(x)
g(x)+ Q′(x)g(x)
f ′(x)g(x) = f (x)g′(x) + Q′(x)[g(x)]2
Q′(x) =f ′(x)g(x)− g′(x)f (x)
[g(x)]2
![Page 89: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/89.jpg)
The Quotient Rule
Let Q(x) = f (x)g(x) .
f (x) = Q(x)g(x)
f ′(x) = Q(x)g′(x) + Q′(x)g(x)
f ′(x) =f (x)g′(x)
g(x)+ Q′(x)g(x)
f ′(x)g(x) = f (x)g′(x) + Q′(x)[g(x)]2
Q′(x) =f ′(x)g(x)− g′(x)f (x)
[g(x)]2
![Page 90: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/90.jpg)
The Quotient Rule
The Quotient Rule
ddx
f (x)
g(x)=
f ′(x)g(x)− g′(x)f (x)
[g(x)]2
Alternate DefinitionLet u = f (x) and v = g(x). Then
ddx
(uv
)=
vu′ − uv′
v2
![Page 91: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/91.jpg)
The Quotient Rule
The Quotient Rule
ddx
f (x)
g(x)=
f ′(x)g(x)− g′(x)f (x)
[g(x)]2
Alternate DefinitionLet u = f (x) and v = g(x). Then
ddx
(uv
)=
vu′ − uv′
v2
![Page 92: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/92.jpg)
Examples
Example
Find the equation of the tangent line to f (x) = x+1x at the point where
x = 3.
f ′(x) =x d
dx(x + 1)− (x + 1) ddx(x)
x2
=x(1)− (x + 1)(1)
x2
=−1x2
∣∣∣∣x=3
= −19
![Page 93: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/93.jpg)
Examples
Example
Find the equation of the tangent line to f (x) = x+1x at the point where
x = 3.
f ′(x) =
x ddx(x + 1)− (x + 1) d
dx(x)
x2
=x(1)− (x + 1)(1)
x2
=−1x2
∣∣∣∣x=3
= −19
![Page 94: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/94.jpg)
Examples
Example
Find the equation of the tangent line to f (x) = x+1x at the point where
x = 3.
f ′(x) =x d
dx(x + 1)− (x + 1) ddx(x)
x2
=x(1)− (x + 1)(1)
x2
=−1x2
∣∣∣∣x=3
= −19
![Page 95: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/95.jpg)
Examples
Example
Find the equation of the tangent line to f (x) = x+1x at the point where
x = 3.
f ′(x) =x d
dx(x + 1)− (x + 1) ddx(x)
x2
=x(1)− (x + 1)(1)
x2
=−1x2
∣∣∣∣x=3
= −19
![Page 96: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/96.jpg)
Examples
Example
Find the equation of the tangent line to f (x) = x+1x at the point where
x = 3.
f ′(x) =x d
dx(x + 1)− (x + 1) ddx(x)
x2
=x(1)− (x + 1)(1)
x2
=−1x2
∣∣∣∣x=3
= −19
![Page 97: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/97.jpg)
Examples
Example
Find the equation of the tangent line to f (x) = x+1x at the point where
x = 3.
f ′(x) =x d
dx(x + 1)− (x + 1) ddx(x)
x2
=x(1)− (x + 1)(1)
x2
=−1x2
∣∣∣∣x=3
= −19
![Page 98: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/98.jpg)
Examples
What else do we need?
y∣∣∣∣x=3
=
43
y = mx + b43
= −19
(3) + b
43
= −13
+ b
53
= b
And the tangent line is y = − 19 x + 5
3 .
![Page 99: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/99.jpg)
Examples
What else do we need?
y∣∣∣∣x=3
=43
y = mx + b43
= −19
(3) + b
43
= −13
+ b
53
= b
And the tangent line is y = − 19 x + 5
3 .
![Page 100: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/100.jpg)
Examples
What else do we need?
y∣∣∣∣x=3
=43
y = mx + b43
= −19
(3) + b
43
= −13
+ b
53
= b
And the tangent line is y = − 19 x + 5
3 .
![Page 101: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/101.jpg)
Examples
What else do we need?
y∣∣∣∣x=3
=43
y = mx + b43
= −19
(3) + b
43
= −13
+ b
53
= b
And the tangent line is y = − 19 x + 5
3 .
![Page 102: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/102.jpg)
Normal Lines
Example
Find the equation of the normal line to the curve f (x) = x+1x at the
point x = 3.
Anyone know what is a normal line?
DefinitionThe normal line to a curve at a point is the line perpendicular to thatcurve at the point. That is, it is the line perpendicular to the tangentline at the point of tangency.
![Page 103: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/103.jpg)
Normal Lines
Example
Find the equation of the normal line to the curve f (x) = x+1x at the
point x = 3.
Anyone know what is a normal line?
DefinitionThe normal line to a curve at a point is the line perpendicular to thatcurve at the point. That is, it is the line perpendicular to the tangentline at the point of tangency.
![Page 104: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/104.jpg)
Normal Lines
Example
Find the equation of the normal line to the curve f (x) = x+1x at the
point x = 3.
Anyone know what is a normal line?
DefinitionThe normal line to a curve at a point is the line perpendicular to thatcurve at the point. That is, it is the line perpendicular to the tangentline at the point of tangency.
![Page 105: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/105.jpg)
Normal Lines
x
y
![Page 106: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/106.jpg)
Normal Lines
x
y
![Page 107: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/107.jpg)
Normal Lines
x
y
![Page 108: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/108.jpg)
Normal Lines
x
y
![Page 109: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/109.jpg)
Normal Lines
If the tangent line has a slope of −19 at the point where x = 3, what is
the slope of the normal line at the same point?
m = 9
y = mx + b43
= 9(3) + b
43
= 27 + b
− 773
= b
And so the equation of the normal line is y = 9x− 773 .
![Page 110: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/110.jpg)
Normal Lines
If the tangent line has a slope of −19 at the point where x = 3, what is
the slope of the normal line at the same point? m = 9
y = mx + b43
= 9(3) + b
43
= 27 + b
− 773
= b
And so the equation of the normal line is y = 9x− 773 .
![Page 111: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/111.jpg)
Normal Lines
If the tangent line has a slope of −19 at the point where x = 3, what is
the slope of the normal line at the same point? m = 9
y = mx + b
43
= 9(3) + b
43
= 27 + b
− 773
= b
And so the equation of the normal line is y = 9x− 773 .
![Page 112: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/112.jpg)
Normal Lines
If the tangent line has a slope of −19 at the point where x = 3, what is
the slope of the normal line at the same point? m = 9
y = mx + b43
= 9(3) + b
43
= 27 + b
− 773
= b
And so the equation of the normal line is y = 9x− 773 .
![Page 113: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/113.jpg)
Normal Lines
If the tangent line has a slope of −19 at the point where x = 3, what is
the slope of the normal line at the same point? m = 9
y = mx + b43
= 9(3) + b
43
= 27 + b
− 773
= b
And so the equation of the normal line is y = 9x− 773 .
![Page 114: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/114.jpg)
Normal Lines
If the tangent line has a slope of −19 at the point where x = 3, what is
the slope of the normal line at the same point? m = 9
y = mx + b43
= 9(3) + b
43
= 27 + b
− 773
= b
And so the equation of the normal line is y = 9x− 773 .
![Page 115: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/115.jpg)
Higher Order Derivatives
What do you think it means to take the 2nd derivative?
Example
Find d2
dx2 (3x4 − 2x2).
d2
dx2 (3x4 − 2x2) =ddx
(12x3 − 4x)
= 36x2 − 4
![Page 116: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/116.jpg)
Higher Order Derivatives
What do you think it means to take the 2nd derivative?
Example
Find d2
dx2 (3x4 − 2x2).
d2
dx2 (3x4 − 2x2) =ddx
(12x3 − 4x)
= 36x2 − 4
![Page 117: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/117.jpg)
Higher Order Derivatives
What do you think it means to take the 2nd derivative?
Example
Find d2
dx2 (3x4 − 2x2).
d2
dx2 (3x4 − 2x2) =
ddx
(12x3 − 4x)
= 36x2 − 4
![Page 118: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/118.jpg)
Higher Order Derivatives
What do you think it means to take the 2nd derivative?
Example
Find d2
dx2 (3x4 − 2x2).
d2
dx2 (3x4 − 2x2) =ddx
(12x3 − 4x)
= 36x2 − 4
![Page 119: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/119.jpg)
Higher Order Derivatives
What do you think it means to take the 2nd derivative?
Example
Find d2
dx2 (3x4 − 2x2).
d2
dx2 (3x4 − 2x2) =ddx
(12x3 − 4x)
= 36x2 − 4
![Page 120: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/120.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 121: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/121.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 122: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/122.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) =
3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 123: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/123.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 124: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/124.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) =
6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 125: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/125.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 126: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/126.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) =
6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 127: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/127.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 128: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/128.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) =
0
f (k)(x) for all k > 4 is 0
![Page 129: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/129.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0
![Page 130: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/130.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is
0
![Page 131: 3.3 Differentiation Rules - btravers.weebly.combtravers.weebly.com/uploads/6/7/2/9/6729909/3.3_differentiation_rul… · x3.3 Differentiation Rules. Constant Functions What does a](https://reader033.fdocuments.us/reader033/viewer/2022042712/5f9b59866c2b6b0ae8432c85/html5/thumbnails/131.jpg)
Higher Order Derivatives
Example
Find derivatives of all orders for f (x) = x3 + 3x + 1.
f (0)(x) = x3 + 3x + 1
f ′(x) = 3x2 + 3
f ′′(x) = 6x
f (3)(x) = 6
f (4)(x) = 0
f (k)(x) for all k > 4 is 0