Q. 7 - Calculus

Question

Explain how the formula for differentiating the natural logarithm function is a special case of the formula for differentiating logarithmic functions of the form ${\log }_{b}x$

Step-by-Step Solution

Answer

The reason has been explained.

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 form ${\log }_{b}x$.

Step 2. Explanation

For any constant b>0 with b$\ne$1 and all approximate values of x,

$\frac{d}{dx}\left({\log }_{b}x\right)=\frac{1}{x\ln b},\frac{d}{dx}\ln x=\frac{1}{x},\frac{d}{dx}\ln \left|x\right|=\frac{1}{x}$

The second rule is a special case of the first with b = e. Because In x has domain (0,$\infty$) when we say that $\frac{d}{dx}\ln x=\frac{1}{x}$ we are also restricting $\frac{1}{x}$ to the domain (0,$\infty$).

In the third rule we consider the full domain as (-$\infty$,0)$\cup$(0,$\infty$).