[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] Chabot Mathematics §2.4 Derivative Chain Rule
Dec 30, 2015
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics§2.4
Derivative
Chain Rule
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §2.3 → Product & Quotient Rules
Any QUESTIONS About HomeWork• §2.3 → HW-9
2.3
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Bruce Mayer, PE Chabot College Mathematics
§2.4 Learning Goals
Define the Chain Rule
Use the chain rule to find and apply derivatives
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Bruce Mayer, PE Chabot College Mathematics
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:
xgxgfxf
uxg
xgufxf
'''
:usingor
'''
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Bruce Mayer, PE Chabot College Mathematics
The Chain Rule - Stated
That is, the derivative of the composite function is the derivative of the “outside” function times the derivative of the “inside” function.
xfxgxgfxgufxf ''''''
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Bruce Mayer, PE Chabot College Mathematics
Chain Rule – Differential Notation
A Simpler, but slightly Less Accurate, Statement of the Chain Rule →
If y = f(u) and u = g(x), then:
• Again Approximating the differentials as algebraic quantities arrive at “Differential Cancellation” which helps to Remember the form of the Chain Rule
dx
du
du
dy
dx
dy
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx7
Bruce Mayer, PE Chabot College Mathematics
Chain Rule Demonstrated
Without chain rule, using expansion:
Using the Chain Rule:
4814412 22 xxxdx
dx
dx
d
uuxdx
du
xdx
du
du
du
dx
dx
dx
d
422122
1212 222
212:Let uyxu
481244 xxu
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Bruce Mayer, PE Chabot College Mathematics
ChainRule Proof
Do OnWhiteBoard
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Bruce Mayer, PE Chabot College Mathematics
Example Chain Ruling Given:
Then Find:
SOLUTION Since y is a function
of x and x is a function of t, can use Chain Rule
By Chain Rule
• Sub x = 1−3t
txxxy 31&3
0
0 @
tdt
dyt
dt
dy
313 2 x
13133 2 t
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx10
Bruce Mayer, PE Chabot College Mathematics
Example Chain Ruling Thus
Then when t = 0
Soif
Then finally
13133 2 tdt
dy
103133 2
0
dt
dy
1133 2
0
dt
dy
61330
tdt
dy
tx
xxy
31
&
3
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Bruce Mayer, PE Chabot College Mathematics
The General Power Rule
If f(x) is a differentiable function, and n is a constant, then
The General Power Rule can be proved by combining the PolyNomial-Power Rule with the Chain Rule• Students should do the proof ThemSelves
xfnxfnnxfdx
d'1
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Bruce Mayer, PE Chabot College Mathematics
Example General Pwr Rule
Find
11233 2322 xdx
dxu
dx
du
du
d
32 1
12
x
x
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx13
Bruce Mayer, PE Chabot College Mathematics
Example Productivity RoC
The productivity, in Units per week, for a sophisticated engineered product is modeled by:
• Where w ≡ The Prouciton-Line Labor Input in Worker-Days per Unit Produced
At what rate is productivity changing when 5 Worker-Days are dedicated to production?
wwwP 303 2
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Bruce Mayer, PE Chabot College Mathematics
Example Productivity RoC
SOLUTION Need to find: First Find the general Derivative of the
Productivity Function. Note
that: P(w) is now in form of [f(x)]n → Use the
General Power Rule
5wdwdP
2/122 303303 wwwwwP
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Bruce Mayer, PE Chabot College Mathematics
Example Productivity RoC
Employing the General Power Rule
2/12 303' wwdw
dwP
wwdw
dww 303303
2
1 212
12
ww
w
3032
3062
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Bruce Mayer, PE Chabot College Mathematics
Example Productivity RoC
So when w = 5 WrkrDays
STATE: So when labor is 5 worker-days, productivity is increasing at a rate of 2 units/week per additional worker-day; i.e., 2 units/[week·WrkrDay].
3056530532
15'
2/12
5
wdw
dPP
22252
60303015075
2
1 2/1
5
wdw
dP
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx17
Bruce Mayer, PE Chabot College Mathematics
Example Productivity RoC
0 1 2 3 4 5 6 7 80
2
4
6
8
10
12
14
16
18
20
w (WorkerHours)
P (
Un
its/W
ee
k)MTH15 • Productivity Sensitivity
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx18
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 06Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = 0; xmax = 8; ymin =0; ymax = 20;% The FUNCTIONx = linspace(xmin,xmax,500); y1 = sqrt(3*x.^2+30*x); y2 = 2*(x-5) + 15% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onplot(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
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx19
Bruce Mayer, PE Chabot College Mathematics
Example Productivity RoC
Check Extremes for very large w
• At Large w, P is LINEAR
The Productivity Sensitivity
• Note that this consist with the Productivity
wwwwwPww
33303limlim 22
33
3
32
6
3032
306limlim
22
w
w
w
w
ww
w
dw
dPww
33lim3lim
wdw
dwP
dw
d
dw
dPww
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx20
Bruce Mayer, PE Chabot College Mathematics
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
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx21
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
DynamicSystemAnalogy
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx25
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx26
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx27
Bruce Mayer, PE Chabot College Mathematics
ChainRule Proof Reference
D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth Publishing Co., 1974, ISBN 0-534-00301-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
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Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx29
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx30
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx31
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx32
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx33
Bruce Mayer, PE Chabot College Mathematics
MuPAD Code
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Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx35
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx36
Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx38
Bruce Mayer, PE Chabot College Mathematics
Mu
PA
D C
od
e
Bruce Mayer, PEMTH15 06Jul13P2.4-76
dEdv := 2*k*(v-35)/v - (k*(v-35)^2+22)/v^2dEdvS := Simplify(dEdv)dEdvN := subs(dEdvS, k = 0.074)U := (w-35)^2expand(U)
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics