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Chapter 2: Q. 7 (page 221)

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

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 formlogb⁡x.

02

Step 2. Explanation

For any constant b>0 with b≠1 and all approximate values of x,

ddxlogb⁡x=1xln⁡b,ddxln⁡x=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 ddxln⁡x=1xwe are also restricting 1xto the domain (0,∞).

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

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