1. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.4 Derivative Chain Rule

2. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 2 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

3. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §2.4 Learning Goals  Define the Chain Rule  Use the chain rule to find and apply derivatives

4. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 4 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:

5. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 5 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.

6. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 6 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

7. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 7 Bruce Mayer, PE Chabot College Mathematics Chain Rule Demonstrated  Without chain rule, using expansion:  Using the Chain Rule:

8. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 8 Bruce Mayer, PE Chabot College Mathematics ChainRule Proof Do On White Board

9. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 9 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

10. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Chain Ruling  Thus  Then when t = 0  So if  Then finally

11. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 11 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

12. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  General Pwr Rule  Find

13. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 13 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?

14. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 14 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

15. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Productivity RoC  Employing the General Power Rule

16. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 16 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].

17. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Productivity RoC

18. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 18 Bruce Mayer, PE Chabot College Mathematics MATLAB Code 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

19. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 19 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

20. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 20 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

21. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 21 Bruce Mayer, PE Chabot College Mathematics All Done for Today Dynamic System Analogy

22. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 22 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

23. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 23 Bruce Mayer, PE Chabot College Mathematics

24. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 24 Bruce Mayer, PE Chabot College Mathematics

25. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 25 Bruce Mayer, PE Chabot College Mathematics

26. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 26 Bruce Mayer, PE Chabot College Mathematics

27. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 27 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

28. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 28 Bruce Mayer, PE Chabot College Mathematics

29. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 29 Bruce Mayer, PE Chabot College Mathematics

30. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 30 Bruce Mayer, PE Chabot College Mathematics

31. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 31 Bruce Mayer, PE Chabot College Mathematics

32. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 32 Bruce Mayer, PE Chabot College Mathematics

33. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 33 Bruce Mayer, PE Chabot College Mathematics MuPAD Code

34. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 34 Bruce Mayer, PE Chabot College Mathematics

35. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 35 Bruce Mayer, PE Chabot College Mathematics

36. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 36 Bruce Mayer, PE Chabot College Mathematics

37. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 37 Bruce Mayer, PE Chabot College Mathematics

38. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 38 Bruce Mayer, PE Chabot College Mathematics 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)

39. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 39 Bruce Mayer, PE Chabot College Mathematics

40. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 40 Bruce Mayer, PE Chabot College Mathematics

41. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 41 Bruce Mayer, PE Chabot College Mathematics

42. BMayer@ChabotCollege.edu MTH15_Lec-09_sec_2-4_Derivative_Chain_Rule_.pptx 42 Bruce Mayer, PE Chabot College Mathematics