Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §4.3 Exp & Log Derivatives Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 4.2 Review § • Any QUESTIONS About • §4.2 → Logarithmic Functions • Any QUESTIONS About HomeWork • §4.2 → HW-19

3. §4.3 Learning Goals • Differentiate exponential and logarithmic functions • Examine applications involving exponential and logarithmic derivatives • Employ logarithmic differentiation

4. Derivative of ex • For any Real Number, x • Thus the exfcn has the unusual property that the derivative of the fcn is the ORIGINAL fcn • The proof of this is quite complicated. For our purposes we treat this as a formula • For a good proof (in Appendix) see: • D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331

5. Derivative of ex • Using the “repeating” nature of d(ex)/dx • Meaning of Above: for any x-value, say x = 1.9, All of these y-related quantities are equal at e1.9 = 6.686 • The yCoOrd: • The Slope: • The ConCavity:

6. Example  ex Derivative • Differentiate: • Using Rules • Product • Power • ex

7. Chain Rule for eu(x) • If u(x) is a differentiable function of x then • Using the ex derivative property

8. Example  Tangent Line • Find the equation of the tangent line at x = 0 for the function: • SOLUTION: • Use the Point-Slope Line Eqn, y-yAP = m(x-xAP), with • Anchor Point, (xAP,yAP): • Slope at the Anchor Point:

9. Example  Tangent Line • Find Slope at x = 0 • Let: • Then: • Thus: • And by Chain Rule

10. Example  Tangent Line • Then m at x = 0 • Using m and the Anchor-Point in the Pt-Slope Eqn • Convert Line-Eqnto Slope-Intercept form

11. Example  Tangent Line • Tangent Line at (0,1) Graphically

12. Derivative of ln(x) = loge(x) • For any POSITIVEReal Number, x • Thus the ln(x) fcn has the unusual property that derivative Does NOT produce another Log • The proof of this is quite complicated. For our purposes we treat this as a formula • For a good proof (in Appendix) see: • D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331

13. Example  ln Derivative • Find the Derivative of: • Using Rules • Quotient • Power • ln(x)

14. Chain Rule for ln(u(x)) • If u(x)> 0 is a differentiable function of x then • Using the ln(x) derivative property

15. Derivative of ax & loga(x) • For Base a with a>0 and a≠1, then for ALL x: • For Base a with a>0 and a≠1, then for ALL x>0: • Prove Both on White/Black Board

16. Example  Revenue RoC • The total number of hits (in thousands) to a website t months after the beginning of 1996 is modeled by • The Model for the weekly advertising revenue in ¢ per hit: • Use the Math Models to determine the daily revenue change at the beginning of the year 2005

17. Example  Revenue RoC • SOLUTION: • The rate of change in Total Revenue, R(t), is the Derivative of the Product of revenue per hit and total hits:

18. Example  Revenue RoC • Thus • Next find t in months for 1996→2005 • Then the rate derivative at t = 108 mon • A units analysis

19. Example  Revenue RoC • The units on H are kHits, and units on r are ¢/Hit. The units on time were months so the derivative has units k¢/mon. Convert to $/mon: • STATE: at the beginning of 2005 the website was making about $690.13 LESS each month that passed.

20. Helpful Hint  Log Diff • Logarithmic Differentiation • Some derivatives are easier to calculate by • first take the natural logarithm of the expression • Next judiciously use the log rules • then take the derivative of both sides of the equation • finally solve for the derivative term

21. Example  Using Log Diff • Using logarithmic differentiation to find the df/dx for: • SOLUTION: • Computing the derivative directly would involve the repeated use of the product rule (not impossible, but very tedious) • Instead, use properties of logarithms to first expand the expression

22. Example  Using Log Diff • Let y = f(x) → • Then take the natural logarithm of both sides: • Use the Power & Log Rules • Now Take the Derivative of Both Sides

23. Example  Using Log Diff • By the Chain Rule • Then • Or • This is a form of Implicit Differentiation; Need to algebraically Isolate dy/dx

24. Example  Using Log Diff • Solving for dy/dx • Recall • Thus • This result would have much more difficult to obtain without the use of the Log transform and implicit differentiation

25. WhiteBoard Work • Problems From §4.3 • P76 → Per Capita Growth • P90 → Newtons Law of (convective) Cooling • Requires a Biot Number* of Less than 0.1 *B. V. Karlekar, R. M. Desmond, Engineering Heat Transfer, St. Paul, MN, West Publishing Co., 1977, pp. 103-110

26. All Done for Today ForPHYS4AStudents From RigidBody Motion-Mechanics

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

28. ConCavity Sign Chart ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection

29. Summary of Log Rules • For any positive numbers M, N, and a with a≠ 1, and whole number p Product Rule Power Rule Quotient Rule Base-to-Power Rule

30. Change of Base Rule • Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then logbx can be converted to a different base as follows:

31. Derive Change of Base Rule • Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10

32. Prove d(ex)/dx =ex • D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331

33. Prove d(ex)/dx =ex

34. D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 325-331

35. Prove: