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Derivatives of the Natural Logarithm

Derivatives of the Natural Logarithm. Monday, April 28 th. Recall: . A while back we did this investigative activity. First divide: 20/a = b. Then find the exponent a b . Example: If a = 5 Then b = 20/5 = 4. And 5 4 = 625. What value of ‘a’ gives the highest value of a b ?

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Derivatives of the Natural Logarithm

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  1. Derivatives of the Natural Logarithm Monday, April 28th

  2. Recall: A while back we did this investigative activity First divide: 20/a = b. Then find the exponent ab. Example: If a = 5 Then b = 20/5 = 4. And 54 = 625. What value of ‘a’ gives the highest value of ab? Note: ‘a’ can be a decimal.

  3. Investigative Activity This is the graph of f(a) = a20/a

  4. Investigative Activity This is the graph of f(a) = a20/a with it’s derivative: f’(a) = 20a20/a–2(1 – ln(a)) e

  5. Derivatives of the Natural Logarithm • This derivative required (among other things) taking the derivative of ln(x). f’(a) = 20a20/a–2(1 – ln(a))

  6. Derivative of the Natural Logarithm f(x) = ln(x) Investigative Activity

  7. Derivative of the Natural Logarithm If f(x) = ln(x) 1 x Then f’(x) =

  8. Derivative of the Natural Logarithm If f(x) = 3ln(x), what is f’(x)? • ln(3x) • 1/x • 3x • 3/x

  9. Derivative of the Natural Logarithm If f(x) = 3ln(x), what is f’(x)? • ln(3x) • 1/x • 3x • 3 / x

  10. Derivative of the Natural Logarithm If f(x) = ln(2x3 – x2), what is f’(x)? • 1 / (2x3 – x2) • 1 / (6x2 – 2x) • 6x2– 2x / x • (6x2– 2x) / (2x3 – x2)

  11. Derivative of the Natural Logarithm If f(x) = ln(2x3 – x2), what is f’(x)? • 1 / (2x3 – x2) • 1 / (6x2 – 2x) • 6x2– 2x / x • (6x2– 2x) / (2x3 – x2)

  12. Practice in your teams • Take the derivative of the following: f(x) = ln(3x) f’(x) = 3 / 3x = 1 / x f(x) = 3ln(x) f’(x) = 3 / x f(x) = xln(x) f’(x) = ln(x) + x/x = ln(x) + 1

  13. Review Chapter 5 What new ideas did we learned in this chapter? • Graphing ex and its derivative • Using the natural logarithm to solve exponents • Using product, chain, and quotient rule to take derivatives of exponents with any base (and natural logarithms today) • Proving that the derivative of ex is ex • Building exponential models

  14. Review Game

  15. Review • Graph f(x) = 2x from -2 < x < 2

  16. Review • Graph f(x) = ex from -2 < x < 2

  17. Review • Graph f(x) = (½)x from -2 < x < 2

  18. Review • If f(x) = x2, what is f’(x)?

  19. Review • If f(x) = 2x, what is f’(x)?

  20. Review • Solve for x in the equation: 3ln(4x) – 2 = 13

  21. Review • If f(x) = e3x, what is f’’(x)?

  22. Review • If f’(x) = e2x, what was the function, f(x), that got you this derivative?

  23. Review • If f(x) = ln(sin(x)), what is f’(x)?

  24. Review • If f(x) = ecos(3x), what is f’(x)?

  25. Review • If f(x) = (ex)2 ex, what is f’(x)?

  26. Review • If f(x) = ex sin(½πx) what is f’(x)?

  27. Review • If f(x) = ex ln(x) what is f’(x)?

  28. Review • If f(x) = 2x 3x what is f’(x)?

  29. Review • If f(x) = (x½)(½x) what is f’(x)?

  30. Review • Where is the function f(x) = ex/x at a local maximum/maximum point?

  31. Review • Where is the function f(x) = ex/x at a local minimum/maximum point? f’(x) = ex (x – 1) / x2 0 = ex (x – 1) / x2 x = 1

  32. Review • Does the function f(x) = ex/x have a local maximum or a minimum point? Why?

  33. Review • Does the function f(x) = ex/x have a local maximum or a minimum point? Why? f’(x) = ex (x – 1) / x2 f’(x) = ex/x – ex/x2 f’’(x) = ex(x – 1)/x2 – ex(x2 – 2x)/x4 f’’(1) = e1(1 – 1)/12– e1(12 – 2(1))/14 = +e Therefore, x = 1 is a minimum point.

  34. Review • If f(x) = ex cos2(x) what is f’(x)?

  35. Review If 3ex+3 = 12, what is x?

  36. Review If f(x) = e3x – 2x, at what point is the slope of this function = 1?

  37. Review If f(x) = e3x – 2x, what is the equation of the tangent at x = 0?

  38. Review • Questions on the back of your sheet!

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