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Lesson Plan #17
Class: AP Calculus
Date: Tuesday October 18th, 2011
Topic: The Chain Rule
Aim: How do we use the chain rule to find the derivative of a function?
Objectives:
1) Students will be able to use the chain rule to find the derivative of a function.
HW# 17: Show ALL work!
1) Find the derivative of
2
A) 𝑦 = √3 − 2𝑥
B) 𝑦 = (5𝑥+1)3
C) 𝑦 = √𝑥 2 + 2𝑥 − 1
2) Find an equation of the line tangent to the graph of f at the point ( 2,3) where f ( x ) 
2x  5 .
Do Now:
1) At which point(s) does the graph of
f ( x) 
x2
has a horizontal tangent line.
x2 1
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Suppose we had to find the derivative of
expression to some power, such as
y  x 2  1 . We have a rule to differentiate x to some power, such as 3x 9 , but not an
3( x 2  1) 9
Let’s examine the function y  2(3 x  5)
First find
dy
, which means the derivative of y with respect to x?
dx
Now let (3 x  5) =
So we have y  2u
Next find
dy
du
Next find
du
dx
u
2
Notice that
dy dy du


dx du dx
This known as the chain rule to find the derivative of a composite function.
You can think of the chain rule like this
If
y  c(u ) n , where u is some expression, then y'  n  c(u) n1  u'
Another way to think of it is that it is like the power rule, except that you have to then multiply by the derivative of the inside.
For each of the following, find
1)
y  (3x 4  4 x 2  8) 6
2)
y  t 2  2t  1
3) y 
3
3x 3  4 x
9
(2t  8) 6
4)
y
5)
f (t )  (3x 2  5) 3
6)
y
2
3
2x  1
dy
.
dx
3
7)
y  2( x  1)
8)
f ( x)  x(3x  9) 3
2
3
Sample Test Questions:
1) If f ( x)  3 x  2  , then find f ' ( x ) .
5
dy
2) If y  3  x  x , then find dx
2
5
3) If y 
(1  x 2 )3
4) If y 
A)
3  2 x , find
1
2 3  2x
5) If
y
, then find
dy
dx
dy
dx
B)

C)
3  2 x  2

C)

3
1
3  2x
3
D)

1
3  2x
D)

4
10
5 x  1 3
3
E)
dy
2
, find
3
dx
5 x  1
30
5 x  12
A)

E)
30
5 x  14
B)  305 x  1
4
6
5 x  14
3
2
3  2 x  2
3