§2.4 Derivative Chain Rule Chabot Mathematics §2.4 Derivative Chain Rule Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

2.3 Review § Any QUESTIONS About Any QUESTIONS About HomeWork §2.3 → Product & Quotient Rules Any QUESTIONS About HomeWork §2.3 → HW-9

§2.4 Learning Goals Define the Chain Rule Use the chain rule to find and apply derivatives

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:

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.

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

Chain Rule Demonstrated Without chain rule, using expansion: Using the Chain Rule:

ChainRule Proof Do On White Board

Example  Chain Ruling Given: Then Find: SOLUTION Since y is a function of x and x is a function of t, can use the Chain Rule By Chain Rule Sub x = 1−3t

Example  Chain Ruling Thus Then when t = 0 So if Then finally

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

Example  General Pwr Rule Find

Example  Productivity RoC The productivity, in Units per week, for a sophisticated engineered product is modeled by: 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?

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

Example  Productivity RoC Employing the General Power Rule = 𝑑 𝑑𝑢 𝑢 1 2 ∙ 𝑑𝑢 𝑑𝑤

Example  Productivity RoC So when w = 5 WrkrDays STATE: So when the 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].

Example  Productivity RoC

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 MATLAB Code

Example  Productivity RoC Check Extremes for very large w At Large w, P is LINEAR The Productivity Sensitivity Note that this is consistent with Productivity

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

Dynamic System Analogy All Done for Today Dynamic System Analogy http://www.webelements.com/webelements/elements/text/Ag/xtal.html

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –

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

MuPAD Code

MuPAD Code Bruce Mayer, PE MTH15 06Jul13 P2.4-76 dEdv := 2*k*(v-35)/v - (k*(v-35)^2+22)/v^2 dEdvS := Simplify(dEdv) dEdvN := subs(dEdvS, k = 0.074) U := (w-35)^2 expand(U)