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MTH 251 – Differential Calculus
Chapter 3 – Differentiation
Section 3.6
The Chain Rule
Copyright © 2010 by Ron Wallace, all rights reserved.
Composition of Functions
• How will x turns of
gear A affect gear
C?
 x turns of A will
result in 2x turns of
B
• B(x) = 2x = n
 n turns of B will
result in 1/3 turn of
C
• C(n) = 1/3 n
 therefore …
• C(B(x)) = 1/3(2x)
= 2/3 x
A
B
C
Image: Copyright © 2006 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
Note that …
dB dn

2
dx dx
dC 1

dn 3
dC 2 dC dn
 

dx 3 dn dx
An algebraic example …
• Consider …
y  4 x 2  20 x  25  (2 x  5)2  (u ) 2
Let u = 2x – 5
dy
 2u  2(2 x  5)  4 x  10
du
du
2
dx
dy
dy du
 8 x  20  (4 x  10)2 

dx
du dx
An algebraic example …
with alternate
notation …
• Consider …
f ( x)  4 x 2  20 x  25  (2 x  5)2  (u )2   g ( x )
2
Let u = g(x) = 2x – 5
f '(u )  f '( g ( x ))  2u  2 g ( x )  2(2 x  5)  4 x  10
g '( x )  2
f '( x )  8 x  20  (4 x  10)2  f '( g ( x))  g '( x)
Generalizing these 2 examples …
• The Chain Rule
If y  f ( g ( x)), then
dy d
  f  g ( x)    f '( g ( x))  g '( x)
dx dx
• Or the alternate form …
If y  f (u ) and u  g ( x), then
dy dy du


dx du dx
“Differential Form”
(more in section 3.11)
1 of 2
The Chain Rule Examples
d
 f ( g ( x))  f '( g ( x))  g '( x)
dx
d
( x 3  3x  1)11  
dx
d
sin( x 2  3)  
dx
d 2 x 7
 e  
dx
2 of 2
The Chain Rule Examples
d
 f ( g ( x ))  f '( g ( x )) g '( x)
dx
d 
tan x  

dx
d  1 



dx  2 x  5 
d 3
cos(esec x )  

dx 
Trigonometric Functions w/ Degrees
• Degrees VS Radians
x 

o
• Example:
180
30o 

180
x
 30  =
 radians 

6
 radians 
d 
d    
o

sin x  
sin 
x 


dx
dx   180  
 

   

cos x o
 cos 
x
180
 180  180
 
Trigonometric Functions w/ Degrees
d 

o
o

sin
x

cos
x
 180
dx 
 
 
d
sin x   cos x
dx
About 57.3 red cycles for the single blue cycle.
x-axis: 0o to 360o
y-axis: -1 to 1
One red cycle with
equal x & y scales.