Understanding the Derivative Chain Rule in Mathematics

1. Chabot Mathematics §2.4 DerivativeChain Rule Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 2.3 Review § • Any QUESTIONS About • §2.3 → Product & Quotient Rules • Any QUESTIONS About HomeWork • §2.3 → HW-9

3. §2.4 Learning Goals • Define the Chain Rule • Use the chain rule to find and apply derivatives

4. 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:

5. 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.

6. 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

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

8. ChainRule Proof Do OnWhiteBoard

9. 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

10. Example  Chain Ruling • Thus • Then when t = 0 • Soif • Then finally

11. 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

12. Example  General Pwr Rule • Find

13. 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?

14. Example  Productivity RoC • SOLUTION • Need to find: • First Find the general Derivative of the Productivity Function. • Notethat: • P(w) is now in form of [f(x)]n → Use the General Power Rule

15. Example  Productivity RoC • Employing the General Power Rule

16. 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].

17. Example  Productivity RoC

18. % 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % 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([xminxmaxyminymax]),... 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

19. 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

20. 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

21. All Done for Today DynamicSystemAnalogy

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

27. 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

33. MuPAD Code

38. 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)