Chabot Mathematics
§2.4 Derivative
Chain Rule
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Review §
2.3
 Any QUESTIONS About
• §2.3 → Product & Quotient Rules
 Any QUESTIONS About
HomeWork
• §2.3 → HW-9
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
§2.4 Learning Goals
 Define the
Chain Rule
 Use the chain
rule to find
and apply
derivatives
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
The Chain Rule
 If y = f(u) is a Differentiable Function of
u, and u = g(x) is a Differentiable
Function of x, then the Composition
Function y = f(g(x)) is also a
Differentiable Function of x whose
Derivative is Given by:
f '  x   f ' u   g '  x 
or using : g  x   u
f '  x   f '  g  x   g '  x 
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
The Chain Rule - Stated
f ' x  f ' u  g ' x  f ' g x g ' x  f ' x
 That is, the derivative of the
composite function is the
derivative of the “outside”
function times the derivative of
the “inside” function.
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chain Rule – Differential Notation
 A Simpler, but slightly Less Accurate,
Statement of the Chain Rule →
 If y = f(u)
and u = g(x),
then:
dy dy du


dx du dx
• Again Approximating the differentials as
algebraic quantities arrive at “Differential
Cancellation” which helps to Remember
the form of the Chain Rule
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chain Rule Demonstrated
 Without chain rule, using expansion:
d
d
2
2 x  1  4 x 2  4 x  1  8 x  4
dx
dx
 Using the Chain Rule:
2
Let : u  2 x  1 y  u
d
d 2
d 2 d
2
2 x  1  u  u  u
dx
dx
du
dx
d
 2u  2 x  1  2u  2  4u
dx
 4u  42 x  1  8 x  4


Chabot College Mathematics
7



 

 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
ChainRule Proof
Do On
White
Board
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Example  Chain Ruling
 Given:
y  x3  x & x  1  3t
 Then Find:
dy
dy
@t  0
dt
dt t 0
 SOLUTION
 Since y is a function
of x and x is a
function of t, can
use the Chain Rule
Chabot College Mathematics
9
 By Chain Rule
dy dy dx
= ×
dt dx dt
d 3
d
= ( x - x ) × (1- 3t )
dx
dt


 3x   1   3
2
• Sub x = 1−3t


 3 31  3t   1
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Example  Chain Ruling
 Thus

 So
if

dy
2
 3 31  3t   1
dt
 Then when t = 0


dy
2
 3 31  3  0  1
dt 0

y  x3  x
&
x  1  3t
 Then finally
dy
 33  1  6
dt t 0

dy
2
 3 31  1
dt 0
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
The General Power Rule
 If f(x) is a differentiable function, and n
is a constant, then
d 
n
n

1


f x   n f x 
 f ' x 

dx 
 The General Power Rule can be
proved by combining the PolyNomialPower Rule with the Chain Rule
• Students should do the proof ThemSelves
Chabot College Mathematics
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Example  General Pwr Rule
é
ù
-2 ù
dê 3 ú dé 2
= ê3( x +1) ú
2
û
dx ê ( x 2 +1) ú dx ë
ë
û
3
d 2 d
d 2
2
 3 u  u  3  2 x 1 
x 1
du
dx
dx
 Find




= -6 ( x +1) × 2x
-3
2

Chabot College Mathematics
12
 12 x
x
2

1
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx


Example  Productivity RoC
 The productivity, in Units per week, for a
sophisticated engineered product is
modeled by:
2
Pw  3w  30w
• Where w ≡ The Production-Line Labor
Input in Worker-Days per Unit Produced
 At what rate would productivity change
when currently 5 Worker-Days are
dedicated to production?
Chabot College Mathematics
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Example  Productivity RoC
 SOLUTION
 Need to find: dP dw w5
 First Find the general Derivative of the
Productivity Function.
 Note
1/ 2
2
2


P
w

3
w

30
w

3
w

30
w
that:
 P(w) is now in form of [f(x)]n → Use the
General Power Rule



Chabot College Mathematics
14

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Example  Productivity RoC
 Employing the General Power Rule
𝑑 1 2 𝑑𝑢
1/ 2
d
2
=
𝑢
∙
P ' w 
3w  30 w
𝑑𝑢
𝑑𝑤
dw



Chabot College Mathematics
15



1
1
1
d
2
2
2
 3w  30 w 
3w  30 w
2
dw
-1/2
1
2
= (3w + 30w) × ( 6w + 30)
2

6 w  30 

2 3w2  30 w
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx

Example  Productivity RoC
 So when w = 5 WrkrDays
1 / 2
dP
1
2
P' 5 
 35  305
 65  30 
dw w5 2


dP
1
60
1/ 2
 75  150  30  30 
2
dw w5 2
2 225
 STATE: So when the labor is 5 workerdays, productivity is increasing at a rate
of 2 units/week per additional workerday; i.e., 2 units/[week·WrkrDay].
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Example  Productivity RoC
MTH15 • Productivity Sensitivity
20
18
P (Units/Week)
16
14
12
10
8
6
4
2
0
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
Chabot College Mathematics
17
1
2
3
4
5
w (WorkerHours)
6
7
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 06Jul13
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
%
% The Limits
xmin = 0; xmax = 8;
ymin =0; ymax = 20;
% The FUNCTION
x = linspace(xmin,xmax,500); y1 = sqrt(3*x.^2+30*x); y2 = 2*(x-5) +
15
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y1, 'LineWidth', 4),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}w (WorkerHours)'),
ylabel('\fontsize{14}P (Units/Week)'),...
title(['\fontsize{16}MTH15 • Productivity Sensitivity',]),...
annotation('textbox',[.5 .05 .0 .1], 'FitBoxToText', 'on',
'EdgeColor', 'none', 'String',
'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)
hold on
plot(x,y2, '-- m', 5,15, 'd r', 'MarkerSize', 10,'MarkerFaceColor',
'r', 'LineWidth', 2)
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:2:ymax])
hold off
Example  Productivity RoC
 Check Extremes for very large w
lim Pw  lim 3w  30w  3w  3  w
2
w
2
w
• At Large w, P is LINEAR
 The Productivity Sensitivity
dP
6 w  30
6w
3w
lim
 lim


 3
w dw
w 
3w
2 3w2  30w
2 3w2
• Note that this is consistent with Productivity
dP
d
d
lim
 3
lim Pw 
3w  3
w dw
dw w
dw

Chabot College Mathematics
19



Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
WhiteBoard Work
 Problems From §2.4
• P74 → Machine Depreciation
• P76 → Specific Power for the
Australian Parakeet (the Budgerigar)
• P80 → Learning Curve
Philip E. Hicks, Industrial
Engineering and
Management: A New
Perspective, McGraw Hill
Publishing Co., 1994,
ISBN-13: 978-0070288072
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
All Done for Today
Dynamic
System
Analogy
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
ChainRule Proof Reference
 D. F. Riddle, Calculus and
Analytical Geometry, Belmont,
CA, Wadsworth Publishing
Co., 1974, ISBN 0-53400301-X pp. 74-76
• This is B. Mayer’s Calculus
Text Book Used in 1974 at
Cabrillo College
– Moral of this story → Do NOT Sell
your Technical Reference Books
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
MuPAD Code
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
37
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
38
dEdv := 2*k*(v-35)/v - (k*(v35)^2+22)/v^2
dEdvS := Simplify(dEdv)
dEdvN := subs(dEdvS, k =
0.074)
U := (w-35)^2
Bruce Mayer, PE
expand(U)
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
MuPAD Code
Bruce Mayer, PE
MTH15 06Jul13
P2.4-76
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
40
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx
Chabot College Mathematics
42
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx