First slide

Introduction to limits

Question

$\frac{d}{d x} (| x |) = \frac{|x|}{x} x \neq 0 then find \frac{d}{d x} (\log | x |)$

Moderate

A

1
B

$\frac{1}{x}$
C

x
D

none

Solution

$\begin{array}{l} x > 0 \frac{d}{d x} | x | = \frac{d}{d x} (x) = 1 = \frac{| x |}{x} \\ x < 0 \frac{d}{d x} | x | = \frac{d}{d x} (- x) = - 1 = \frac{| x |}{x} \\ ∴ \frac{d}{d x} | x | = \frac{| x |}{x} x \neq 0 \\ \frac{d}{d x} (\log | x |) = \frac{1}{| x |} \frac{d}{d x} | x | = \frac{1}{| x |} \frac{| x |}{x} \\ = \frac{1}{x} (x \neq 0) \end{array}$

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