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Derivatives of Logarithmic Functions

Derivatives of Logarithmic Functions. Objective: Obtain derivative formulas for logs. Review Laws of Logs. Algebraic Properties of Logarithms Product Property Quotient Property Power Property Change of base. Review Laws of Logs. Algebraic Properties of Logarithms

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Derivatives of Logarithmic Functions

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  1. Derivatives of Logarithmic Functions Objective: Obtain derivative formulas for logs.

  2. Review Laws of Logs • Algebraic Properties of Logarithms • Product Property • Quotient Property • Power Property • Change of base

  3. Review Laws of Logs • Algebraic Properties of Logarithms • Remember that means .

  4. Review Laws of Logs • Algebraic Properties of Logarithms • Remember that means . • Logarithmic and exponential functions are inverse functions.

  5. Derivatives of Logs • We will start this definition with another way to express e. In chapter 2, we defined e as: • Now, we will look at e as: • We make the substitution v = 1/x, and we know that as

  6. Defintion

  7. Defintion

  8. Defintion

  9. Definition • We will now let v=h/x, so h = vx

  10. Definition • Finally

  11. Defintion • Now we will look at the derivative of a log with any base.

  12. Defintion • Now we will look at the derivative of a log with any base. • We will use the change of base formula to rewrite this as

  13. Defintion • Now we will look at the derivative of a log with any base. • We will use the change of base formula to rewrite this as

  14. Definition • In summary:

  15. Example 1 • The figure below shows the graph of y = lnx and its tangent lines at x = ½, 1, 3, and 5. Find the slopes of the tangent lines.

  16. Example 1 • The figure below shows the graph of y = lnx and its tangent lines at x = ½, 1, 3, and 5. Find the slopes of the tangent lines. • Since the derivative of y = lnx is dy/dx = 1/x, the slopes of the tangent lines are: 2, 1, 1/3, 1/5.

  17. Example 1 • Does the graph of y = lnx have any horizontal tangents?

  18. Example 1 • Does the graph of y = lnx have any horizontal tangents? • The answer is no. 1/x (the derivative) will never equal zero, so there are no horizontal tangent lines. • As the value of x approaches infinity, the slope of the tangent line does approach 0, but never gets there.

  19. Example 2 • Find

  20. Example 2 • Find • We will use a u-substitution and let

  21. Example 3 • Find

  22. Example 3 • Find • We will use our rules of logs to make this a much easier problem.

  23. Example 3 • Now, we solve.

  24. Absolute Value • Lets look at

  25. Absolute Value • Lets look at • If x > 0, |x| = x, so we have

  26. Absolute Value • Lets look at • If x > 0, |x| = x, so we have • If x < 0, |x|= -x, so we have

  27. Absolute Value • Lets look at • If x > 0, |x| = x, so we have • If x < 0, |x|= -x, so we have • So we can say that

  28. Logarithmic Differentiation • This is another method that makes finding the derivative of complicated problems much easier. • Find the derivative of

  29. Logarithmic Differentiation • Find the derivative of • First, take the natural log of both sides and treat it like example 3.

  30. Logarithmic Differentiation • Find the derivative of • First, take the natural log of both sides and treat it like example 3.

  31. Logarithmic Differentiation • Find the derivative of

  32. Homework • Section 3.2 • 1-29 odd • 35, 37

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