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MATH 31 LESSONS
Chapter 2:
Derivatives
4. The Chain Rule
Section 2.6: The Chain Rule
Read Textbook pp. 96 - 102
A. Composite Functions (Review)
A composite function is defined as
where
g (x) is the “inside function”
f is the “outside function”
xgfxgf
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
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
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
f (x) = x2 - 3x + 7
g(x) = 4 - x2
xgfxgf
24 xf
7434 222 xx
7312816 242 xxx
115 24 xx
B. The Chain Rule
For the composite function xgfxgf
xgxgfxgfdx
d
xgxgfxgfdx
d
First, take the derivative of the outside function (and leave the inside function the same) ...
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
The chain rule can also be expressed in Leibnitz notation:
:andIf xguxgfy
dx
du
du
dy
dx
dy
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.
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
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
or
nxfy If
xfxfny n 1
:andIf xguuy n
dx
duuny n 1
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
Method 1: Leibnitz
Let u = x2 - 3x
42 3xxy
Assign u as the “inside function”
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
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
u = x2 - 3x
y = u4
dx
du
du
dy
dx
dy
xxdx
du
du
d324
Substitute y = u4 and u = x2 - 3x
u = x2 - 3x
y = u4
dx
du
du
dy
dx
dy
xxdx
du
du
d324
324 3 xu
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
Method 2: “Outside, Inside”
42 3xxy
The “inside function” is x2 - 3x
42 3xxy
The “inside function” is x2 - 3x
The “outside function” is the 4th power
42 3xxy
First, do the derivative of the outside function.
Be certain to keep the inside function the same
32 34 xxy
42 3xxy
Next, don’t forget to do the derivative of the inside function
xxdx
dxxy 334 232
42 3xxy
xxdx
dxxy 334 232
323432 xxx
Since this method is much faster, we will use this method exclusively from now on.
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
8218
2526
526
xx
xxxf
Bring all the x’s to the top.
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
8218
2526
526
xx
xxxf
21721 5265268 xxdx
dxxxf
Next, don’t forget to do the derivative of the inside function
8218
2526
526
xx
xxxf
21721 5265268 xxdx
dxxxf
32721 251205268 xxxx
32721 1025268 xxxx
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
Express in power notation.
21
2323 7474 tttttg
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
Next, don’t forget to do the derivative of the inside function
21
2323 7474 tttttg
2321
23 74742
1tt
dt
dtttg
21
2323 7474 tttttg
2321
23 74742
1tt
dt
dtttg
tttt 1412742
1 221
23
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
First, do the derivative of the outside function.
Be certain to keep the inside function the same
9424 32 xxxy
8424 329 xxxy
Next, don’t forget to do the derivative of the inside function
9424 32 xxxy
4248424 32329 xxx
dx
dxxxy
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.
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”
4248424 32329 xxx
dx
dxxxy
4248424 32329 xx
dx
dxxx
24324
8424 3342329 xxdx
dxxxxx
xxxxxxx 64342329 33248424
Ex. 6 Find
Try this example on your own first.Then, check out the solution.
3
xdx
dy
2and5if 23 xuuuy
You read this as:
“Find the derivative of y,
and then evaluate it at x = 3”
3
xdx
dy
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
dx
du
du
dy
dx
dy
2123 25 x
dx
duu
du
d
222
1103 2
12 xdx
dxuu
chain rule
222
1103 2
12 xdx
dxuu
122
1103 2
12 xuu
22
103 2
x
uu
Next, evaluate the derivative at x = 3:
2
5 23
xu
uuy
123,3At ux
22
103 2
x
uu
dx
dy
123,3At ux
232
11013 2
3
xdx
dy
2
13
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
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
xgfxgf
xgxgf
Next, don’t forget to do the derivative of the inside function
Now, evaluate the function: g (3) = 6 g (3) = 5
f (5) = 2 f (6) = 8
xgxgfxgf
333 ggfgf
g (3) = 6 g (3) = 5
f (5) = 2 f (6) = 8
xgxgfxgf
333 ggfgf
56 f
g (3) = 6 g (3) = 5
f (5) = 2 f (6) = 8
xgxgfxgf
333 ggfgf
56 f
58 40
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
523 6 xxy
Which rule do you use first?
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
523 6 xxy
523523 66 xxxxy
66563 2423522 xdx
dxxxx
Next, use the chain rule
523 6 xxy
523523 66 xxxxy
66563 2423522 xdx
dxxxx
xxxxx 26563423522
xxxxx 26563423522
424522 61063 xxxx
Put in the same order.
Let A = x2 + 6
xxxxx 26563423522
424522 61063 xxxx
4452 103 AxAx
242 103 xAAx Use substitution to make the factoring easier.
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.
But A = x2 + 6
242 103 xAAx
22422 10636 xxxx
22422 101836 xxxx
Simplify inside the bracket.
But A = x2 + 6
242 103 xAAx
22422 10636 xxxx
22422 101836 xxxx
18136 2422 xxx
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
112
2
1
x
xy
Which rule do you use first?
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
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”.
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:
2
1
2
111
2102
x
x
dx
d
x
xy
2
22102
2
2121
2
111
x
xxxx
x
x
2
2102
2
1122
2
111
x
xxx
x
x
2
2102
2
1122
2
111
x
xxx
x
x
2
22102
2
142
2
111
x
xxx
x
x
2
2102
2
1122
2
111
x
xxx
x
x
2
22102
2
142
2
111
x
xxx
x
x
2
2102
2
14
2
111
x
xx
x
x
2
2102
2
1122
2
111
x
xxx
x
x
2
22102
2
142
2
111
x
xxx
x
x
2
2102
2
14
2
111
x
xx
x
x
12
1022
2
11411
x
xxx This is another possible answer.
