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Explain how the formula for differentiating the natural logarithm function is a special case of the formula for differentiating logarithmic functions of the formlogbx

Short Answer

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Step by step solution

01

Step 1. Given information

We have to explain how the formula for differentiating the natural logarithm function is a special case of the formula for differentiating logarithmic functions of the formlogbx.

02

Step 2. Explanation

For any constant b>0 with b1 and all approximate values of x,

ddxlogbx=1xlnb,ddxlnx=1x,ddxln|x|=1x

The second rule is a special case of the first with b = e. Because In x has domain (0,) when we say that ddxlnx=1xwe are also restricting 1xto the domain (0,).

In the third rule we consider the full domain as (-,0)(0,).

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Most popular questions from this chapter

Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

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f(x)=1x-3x2x5-1x

Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.

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Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park tminutes after she begins her jog is given by the function s(t)shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.

(a) What was the average rate of change of Linda’s distance from the oak tree over the entire thirty-minute jog? What does this mean in real-world terms?

(b) On which ten-minute interval was the average rate of change of Linda’s distance from the oak tree the greatest: the first 10minutes, the second 10minutes, or the last10minutes?

(c) Use the graph of s(t)to estimate Linda’s average velocity during the 5-minute interval fromt=5tot=10. What does the sign of this average velocity tell you in real-world terms?

(d) Approximate the times at which Linda’s (instantaneous) velocity was equal to zero. What is the physical significance of these times?

(e) Approximate the time intervals during Linda’s jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?

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