B.1.7 – Derivatives of Logarithmic Functions Calculus - Santowski 10/8/2015 Calculus - Santowski 1

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  7. Solve 5 x ln5 = 18  8. Solve 3 x+1 = 2 x  9. Sketch y = lnx  10. d/dx e π  11. d/dx x π 10/8/2015 Calculus - Santowski2

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 10/8/2015 Calculus - Santowski3

(A) Derivative Prediction  So, now consider the graph of f(x) = ln(x) and then predict what the derivative graph should look like 10/8/2015 Calculus - Santowski4

(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 10/8/2015 Calculus - Santowski5

(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 10/8/2015 Calculus - Santowski6

(B) Derivatives of Logarithmic Functions  The derivative of the natural logarithmic function is:  And in general, the derivative of any logarithmic function is: 10/8/2015 Calculus - Santowski7

(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 10/8/2015 Calculus - Santowski8

(D) Working with the Derivatives  Differentiate the following: 10/8/2015 Calculus - Santowski9

(E) Working with Tangent Lines  At what point on the graph of y(x) = 3 x + 1 is the tangent line parallel to the line 5x – y – 1 = 0?  At what point on the graph of g(x) = 2e x - 1 is the tangent line perpendicular to 3x + y – 2 = 0? 10/8/2015 Calculus - Santowski10

(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. 10/8/2015 Calculus - Santowski11

(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(x 2 – 1) is increasing and decreasing  5. Find the maximum value of 10/8/2015 Calculus - Santowski12

(G) Internet Links  Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins  Visual Calculus - Derivative of Exponential Function Visual Calculus - Derivative of Exponential Function  From pkving From pkving 10/8/2015 Calculus - Santowski13

HOMEWORK  Text, S4.4, p  (1) Algebra: Q1-27 as needed plus variety  Text, S4.5, p260-1  (1) Algebra: Q1-39 as needed plus variety 10/8/2015 Calculus - Santowski14