ddx(|x|)=xxx≠0 then find ddx(log⁡|x|)

x>0ddx|x|=ddx(x)=1=|x|xx<0ddx|x|=ddx(−x)=−1=|x|x∴ddx|x|=|x|xx≠0ddx(log⁡|x|)=1|x|ddx|x|=1|x||x|x=1x(x≠0)

ddx(|x|)=xxx≠0 then find ddx(log⁡|x|)

x>0ddx|x|=ddx(x)=1=|x|xx<0ddx|x|=ddx(−x)=−1=|x|x∴ddx|x|=|x|xx≠0ddx(log⁡|x|)=1|x|ddx|x|=1|x||x|x=1x(x≠0)

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

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$\frac{1}{x}$

none

answer is B.

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Detailed Solution

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