1. §2.4 Derivative Chain Rule
Chabot Mathematics
§2.4 Derivative Chain Rule
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 2.3 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
§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 On White Board

9. 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

10. Example  Chain Ruling
Thus
Then when t = 0
So if
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 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?

14. 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. Example  Productivity RoC
Employing the General Power Rule
= 𝑑 𝑑𝑢 𝑢 ∙ 𝑑𝑢 𝑑𝑤

16. 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].

17. Example  Productivity RoC

18. 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([ ]); % 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',[ ], '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 is consistent with 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:

21. Dynamic System Analogy
All Done for Today
Dynamic System Analogy

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

27. ChainRule Proof Reference
D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth Publishing Co., 1974, ISBN X pp
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)