[email protected] MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical

[email protected] • MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics§2.4

Derivative

Chain Rule



Review §

Any QUESTIONS About• §2.3 → Product & Quotient Rules

Any QUESTIONS About HomeWork• §2.3 → HW-9

2.3

http://www.google.com/url?sa=i&rct=j&q=higher%20order%20derivatives&source=images&cd=&cad=rja&docid=ZCQuryl0t8jVNM&tbnid=z0a_O_0dC5CH8M:&ved=0CAUQjRw&url=http://education.fcps.org/uhs/node/1386&ei=QT7YUaHTDcXiiAK_u4CACw&bvm=bv.48705608,d.cGE&psig=AFQjCNGVxEQZ5KJCyuR9Tsl6pUMHH9ztoA&ust=1373212608413499



§2.4 Learning Goals

Define the Chain Rule

Use the chain rule to find and apply derivatives

http://www.google.com/url?sa=i&rct=j&q=chain+gang&source=images&cd=&cad=rja&docid=VLa0HBVz3UopkM&tbnid=kfhu1qHmWoYFTM:&ved=0CAUQjRw&url=http://webwanderers.org/2006/08/taylor_pallister_limited.html&ei=SUDYUfnZJ43DiwLqvIDwCQ&psig=AFQjCNEJfGBvmJb5d_5mKYn7fjFt0BhE9Q&ust=1373213018277300



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

'''



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 ''''''



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



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



ChainRule Proof

Do OnWhiteBoard

http://www.google.com/url?sa=i&rct=j&q=whiteBoard&source=images&cd=&cad=rja&docid=fIN_eRoWHj0G0M&tbnid=b1OWxS6TRyGRJM:&ved=0CAUQjRw&url=http://www.beyond.com/articles/timeline-of-training-technology-part-2-1970s-11106-article.html&ei=uyHXUdnbBsLNiwK53oGwBA&bvm=bv.48705608,d.cGE&psig=AFQjCNHffTCfKrbIKlaSGatZAQ7VO-BdVQ&ust=1373139714877810



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



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



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



Example General Pwr Rule

Find

11233 2322 xdx

dxu

dx

du

du

d

32 1

12

x

x



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




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




Employing the General Power Rule

2/12 303' wwdw

dwP

wwdw

dww 303303

2

1 212

12

ww

w

3032

3062




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




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



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




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



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

http://www.google.com/url?sa=i&rct=j&q=budgerigar+bird&source=images&cd=&cad=rja&docid=8BrpXIIXMx-pTM&tbnid=r50qoCvcMQ6e7M:&ved=0CAUQjRw&url=http://true-wildlife.blogspot.com/2011/02/budgerigar.html&ei=_X7YUZjhBKrGigKS_gE&bvm=bv.48705608,d.cGE&psig=AFQjCNHzZM-rdJpj4o02XCvEbWVJcjkNIg&ust=1373229124371294



All Done for Today

DynamicSystemAnalogy

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Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22











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

http://www.google.com/url?sa=i&rct=j&q=isbn+053400301x&source=images&cd=&cad=rja&docid=fKCw8UoJH2KYOM&tbnid=ev8oeMemNvGi0M:&ved=0CAUQjRw&url=http://www.biblio.com/9780534003012&ei=bWPYUc69FqGIiwKrr4CYDA&bvm=bv.48705608,d.cGE&psig=AFQjCNFeUQXfHVRretbnLD_tQrDKc4EB_Q&ust=1373222108795243













MuPAD Code











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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http://www.google.com/url?sa=i&rct=j&q=higher+order+derivatives+worksheet&source=images&cd=&cad=rja&docid=UzYAWOYbX2jrHM&tbnid=eEfIS0ov0gspKM:&ved=0CAUQjRw&url=http://www.hollywoodhighschool.net/apps/pages/index.jsp?uREC_ID=116947&type=u&pREC_ID=210421&ei=m_TWUYr3O8qoigLJ3YHIAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNHYr8PZTh_Skq5LPhpN-8lENIexrA&ust=1373128179068489

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[email protected] MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical