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

B.1.7 – Derivatives of Logarithmic

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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

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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

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(A) Derivative Prediction

So, now consider the graph of f(x) = ln(x) and then predict what the derivative graph should look like

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(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

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(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

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(B) Derivatives of Logarithmic Functions

The derivative of the natural logarithmic function is:

And in general, the derivative of any logarithmic function is:

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d

dx ln(x) =

1

x

€

d

dx loga x =

1

x × lna

(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

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(D) Working with the Derivatives

Differentiate the following:

Differentiate the following:

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y(x) = ln x 2 + 2x −1( )

y(x) = ln1

x

⎛

⎝ ⎜

⎞

⎠ ⎟

y(x) = ln ln x( )

y(x) = ln x( )2

€

f (x) = log2

1

x

⎛

⎝ ⎜

⎞

⎠ ⎟

f (x) = log3 1+ x ln3( )

f (x) = logex

f (x) = x ln x − x

(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?

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(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.

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(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

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f (x) = x − ln x

€

f (x) = x − ln x

€

g(x) = ln x − ex−1

€

h(x) =ln x

x

(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

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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

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