math 31 lessons chapter 2: derivatives 4. the chain rule
TRANSCRIPT
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MATH 31 LESSONS
Chapter 2:
Derivatives
4. The Chain Rule
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Section 2.6: The Chain Rule
Read Textbook pp. 96 - 102
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A. Composite Functions (Review)
A composite function is defined as
where
g (x) is the “inside function”
f is the “outside function”
xgfxgf
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Ex. 1 If f (x) = x2 - 3x + 7 and g(x) = 4 - x2 ,
then find:
Try this example on your own first.Then, check out the solution.
xgf
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f (x) = x2 - 3x + 7
g(x) = 4 - x2
xgfxgf
24 xf
g (x) is the inside function.
Replace it with g (x) = 4 - x2
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f (x) = x2 - 3x + 7
g(x) = 4 - x2
xgfxgf
24 xf
7434 222 xx
Wherever you see x in the f function, replace it with 4 - x2
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f (x) = x2 - 3x + 7
g(x) = 4 - x2
xgfxgf
24 xf
7434 222 xx
7312816 242 xxx
115 24 xx
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B. The Chain Rule
For the composite function xgfxgf
xgxgfxgfdx
d
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xgxgfxgfdx
d
First, take the derivative of the outside function (and leave the inside function the same) ...
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xgxgfxgfdx
d
First, take the derivative of the outside function (and leave the inside function the same) ...
... then, take the derivative of the inside function
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The chain rule can also be expressed in Leibnitz notation:
:andIf xguxgfy
dx
du
du
dy
dx
dy
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dx
du
du
dy
dx
dy
This is easy to remember, because if we treat these as true fractions, the du’s would cancel and you would be left with dy / dx.
But of course, you would never do this.
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The Chain Rule Applied to Power Functions
The most common application of the chain rule
in this unit is when the outside function is a power.
e.g.
y = [ f (x) ] n
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nxfy If
xfxfny n 1
First, take the derivative of the outside power function (and leave the inside function the same) ...
... then, take the derivative of the inside function
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or
nxfy If
xfxfny n 1
:andIf xguuy n
dx
duuny n 1
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Ex. 2 Differentiate using the chain rule.
No need to simplify.
Try this example on your own first.Then, check out the solution.
42 3xxy
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Method 1: Leibnitz
Let u = x2 - 3x
42 3xxy
Assign u as the “inside function”
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Let u = x2 - 3x
Then, y = u4
42 3xxy
When you replace the inside function with u, you are left with just the outside function
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u = x2 - 3x
y = u4
dx
du
du
dy
dx
dy
This is the Leibnitz formula for the chain rule.
Remember, to ensure it is in the proper form, you can “cancel” the du’s and you are left with dy / dx
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u = x2 - 3x
y = u4
dx
du
du
dy
dx
dy
xxdx
du
du
d324
Substitute y = u4 and u = x2 - 3x
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u = x2 - 3x
y = u4
dx
du
du
dy
dx
dy
xxdx
du
du
d324
324 3 xu
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u = x2 - 3x
y = u4
dx
du
du
dy
dx
dy
xxdx
du
du
d324
324 3 xu
3234 2 xxx Back substitute so that the answer is in terms of x
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Method 2: “Outside, Inside”
42 3xxy
The “inside function” is x2 - 3x
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42 3xxy
The “inside function” is x2 - 3x
The “outside function” is the 4th power
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42 3xxy
First, do the derivative of the outside function.
Be certain to keep the inside function the same
32 34 xxy
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42 3xxy
Next, don’t forget to do the derivative of the inside function
xxdx
dxxy 334 232
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42 3xxy
xxdx
dxxy 334 232
323432 xxx
Since this method is much faster, we will use this method exclusively from now on.
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Ex. 3 Differentiate using the chain rule.
No need to simplify.
Try this example on your own first.Then, check out the solution.
8
2
526
xxxf
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8218
2526
526
xx
xxxf
Bring all the x’s to the top.
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8218
2526
526
xx
xxxf
721 5268 xxxf
First, do the derivative of the outside function.
Be certain to keep the inside function the same
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8218
2526
526
xx
xxxf
21721 5265268 xxdx
dxxxf
Next, don’t forget to do the derivative of the inside function
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8218
2526
526
xx
xxxf
21721 5265268 xxdx
dxxxf
32721 251205268 xxxx
32721 1025268 xxxx
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Ex. 4 Differentiate using the chain rule.
No need to simplify.
Try this example on your own first.Then, check out the solution.
23 74 tttg
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Express in power notation.
21
2323 7474 tttttg
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First, do the derivative of the outside function.
Be certain to keep the inside function the same
21
2323 7474 tttttg
21
23 742
1tttg
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Next, don’t forget to do the derivative of the inside function
21
2323 7474 tttttg
2321
23 74742
1tt
dt
dtttg
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21
2323 7474 tttttg
2321
23 74742
1tt
dt
dtttg
tttt 1412742
1 221
23
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Ex. 5 Differentiate using the chain rule.
No need to simplify.
Try this example on your own first.Then, check out the solution.
9424 32 xxxy
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First, do the derivative of the outside function.
Be certain to keep the inside function the same
9424 32 xxxy
8424 329 xxxy
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Next, don’t forget to do the derivative of the inside function
9424 32 xxxy
4248424 32329 xxx
dx
dxxxy
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4248424 32329 xxx
dx
dxxxy
4248424 32329 xx
dx
dxxx
Apply the derivative to each part of the inside function.
You will be required to do the chain rule again.
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4248424 32329 xxx
dx
dxxxy
4248424 32329 xx
dx
dxxx
24324
8424 3342329 xxdx
dxxxxx
Derivative of “outside function”
(leave inside same)
Don’t forget the derivative of the “inside function”
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4248424 32329 xxx
dx
dxxxy
4248424 32329 xx
dx
dxxx
24324
8424 3342329 xxdx
dxxxxx
xxxxxxx 64342329 33248424
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Ex. 6 Find
Try this example on your own first.Then, check out the solution.
3
xdx
dy
2and5if 23 xuuuy
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You read this as:
“Find the derivative of y,
and then evaluate it at x = 3”
3
xdx
dy
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First, find the derivative using the chain rule:
2
5 23
xu
uuy
dx
du
du
dy
dx
dy
2123 25 x
dx
duu
du
d
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dx
du
du
dy
dx
dy
2123 25 x
dx
duu
du
d
222
1103 2
12 xdx
dxuu
chain rule
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222
1103 2
12 xdx
dxuu
122
1103 2
12 xuu
22
103 2
x
uu
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Next, evaluate the derivative at x = 3:
2
5 23
xu
uuy
123,3At ux
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22
103 2
x
uu
dx
dy
123,3At ux
232
11013 2
3
xdx
dy
2
13
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Ex. 7 If g (3) = 6 , g (3) = 5, f (5) = 2 , and f (6) = 8,
then evaluate:
Try this example on your own first.Then, check out the solution.
3gf
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Expand the function first in terms of x :
xgfxgf
xgf
First, do the derivative of the outside function.
Be certain to keep the inside function the same
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xgfxgf
xgxgf
Next, don’t forget to do the derivative of the inside function
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Now, evaluate the function: g (3) = 6 g (3) = 5
f (5) = 2 f (6) = 8
xgxgfxgf
333 ggfgf
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g (3) = 6 g (3) = 5
f (5) = 2 f (6) = 8
xgxgfxgf
333 ggfgf
56 f
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g (3) = 6 g (3) = 5
f (5) = 2 f (6) = 8
xgxgfxgf
333 ggfgf
56 f
58 40
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Ex. 8 Differentiate, using more than one rule.
Fully factor your answer.
Try this example on your own first.Then, check out the solution.
523 6 xxy
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523 6 xxy
Which rule do you use first?
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523 6 xxy
Take the derivative of the first and leave the second
+
Leave the first and take the derivative of the second
(u v) = u v + u v
523523 66 xxxxy
Use the product rule first
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523 6 xxy
523523 66 xxxxy
66563 2423522 xdx
dxxxx
Next, use the chain rule
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523 6 xxy
523523 66 xxxxy
66563 2423522 xdx
dxxxx
xxxxx 26563423522
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xxxxx 26563423522
424522 61063 xxxx
Put in the same order.
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Let A = x2 + 6
xxxxx 26563423522
424522 61063 xxxx
4452 103 AxAx
242 103 xAAx Use substitution to make the factoring easier.
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But A = x2 + 6
242 103 xAAx
22422 10636 xxxx
After factoring, back substitute so that it is in terms of only x.
Be certain to use brackets.
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But A = x2 + 6
242 103 xAAx
22422 10636 xxxx
22422 101836 xxxx
Simplify inside the bracket.
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But A = x2 + 6
242 103 xAAx
22422 10636 xxxx
22422 101836 xxxx
18136 2422 xxx
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Ex. 9 Differentiate, using more than one rule.
Fully simplify your answer.
Try this example on your own first.Then, check out the solution.
112
2
1
x
xy
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112
2
1
x
xy
Which rule do you use first?
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112
2
1
x
xy
102
2
111
x
xy
Chain rule first
First, do the derivative of the outside function.
Be certain to keep the inside function the same
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112
2
1
x
xy
2
1
2
111
2102
x
x
dx
d
x
xy
Don’t forget to do the derivative of the “inside function”.
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2
1
2
111
2102
x
x
dx
d
x
xy Use the
quotient rule
2
22102
2
2121
2
111
x
xxxx
x
x
2v
vuvu
v
u
Quotient Rule:
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xxx This is another possible answer.