Properties of the Derivative
Properties of the Derivative

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

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Review Laws of Logs Algebraic Properties of Logarithms 1.Product Property 2.Quotient Property 3.Power Property 4.Change of base

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

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

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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

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Defintion

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

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Definition Finally

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

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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

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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

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Definition In summary:

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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.

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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.

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

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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.

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Example 2 Find

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

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Example 3 Find

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

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Example 3 Now, we solve.

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Absolute Value Lets look at

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

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

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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

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

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

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

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Logarithmic Differentiation Find the derivative of

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Homework Section 3.2 1-29 odd 35, 37

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