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B.1.7 – Derivatives of Logarithmic Functions

B.1.7 – Derivatives of Logarithmic Functions. Calculus - Santowski. Fast Five. 1. Write log 5 8 in terms of ln Simply using properties of logs & exponents: 2. ln(e tanx ) 3. log 2 (8 x-5 ) 4. 3lnx – ln(3x) + ln(12x 2 ) 5. ln(x 2 - 4) – ln(x + 2). 6. Solve 3 x = 19

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B.1.7 – Derivatives of Logarithmic Functions

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  1. B.1.7 – Derivatives of Logarithmic Functions Calculus - Santowski Calculus - Santowski

  2. Fast Five • 1. Write log5 8 in terms of ln • Simply using properties of logs & exponents: • 2. ln(etanx) • 3. log2(8x-5) • 4. 3lnx – ln(3x) + ln(12x2) • 5. ln(x2 - 4) – ln(x + 2) • 6. Solve 3x = 19 • 7. Solve 5xln5 = 18 • 8. Solve 3x+1 = 2x • 9. Sketch y = lnx • 10. d/dx eπ • 11. d/dx xπ Calculus - Santowski

  3. Lesson Objectives • 1. Predict the appearance of the derivative curve of y = ln(x) • 2. Differentiate equations involving logarithms • 3. Apply derivatives of logarithmic functions to the analysis of functions Calculus - Santowski

  4. (A) Derivative Prediction • So, now consider the graph of f(x) = ln(x) and then predict what the derivative graph should look like Calculus - Santowski

  5. (A) Derivative Prediction • Our log fcn is constantly increasing and has no max/min points • So our derivative graph should be positive & have no x-intercepts Calculus - Santowski

  6. (A) Derivative Prediction • So when we use technology to graph a logarithmic function and its derivative, we see that our prediction is correct • Now let’s verify this graphic predication algebraically Calculus - Santowski

  7. (B) Derivatives of Logarithmic Functions • The derivative of the natural logarithmic function is: • And in general, the derivative of any logarithmic function is: Calculus - Santowski

  8. (C) Proofs of the Derivative • Proving that our equations are in fact the correct derivatives and being able to provide and discuss these derivatives will be an “A” level exercise, should you choose to pursue that Calculus - Santowski

  9. (D) Working with the Derivatives • Differentiate the following: • Differentiate the following: Calculus - Santowski

  10. (E) Working with Tangent Lines • At what point on the graph of y(x) = 3x + 1 is the tangent line parallel to the line 5x – y – 1 = 0? • At what point on the graph of g(x) = 2ex - 1 is the tangent line perpendicular to 3x + y – 2 = 0? Calculus - Santowski

  11. (E) Working with Tangent Lines • 1. Find the equation of the tangent line to y = ln(2x – 1) at x = 1 • 2. A line with slope m passes through the origin and is tangent to y = ln(2x) . What is the value of m? • 3. A line with slope m passes through the origin and is tangent to y = ln(x/3) . Find the x-intercept of the line normal to the curve at this tangency point. Calculus - Santowski

  12. (F) Function Analysis • 1. Find the minimum point of the function • 2. Find the inflection point of • 3. Find the maximum point of • 4. Find where the function y = ln(x2 – 1) is increasing and decreasing • 5. Find the maximum value of Calculus - Santowski

  13. (G) Internet Links • Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins • Visual Calculus - Derivative of Exponential Function • From pkving Calculus - Santowski

  14. HOMEWORK • Text, S4.4, p251-253 • (1) Algebra: Q1-27 as needed plus variety • Text, S4.5, p260-1 • (1) Algebra: Q1-39 as needed plus variety Calculus - Santowski

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