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(D) Given $f(x) = |x|.$ We check the differentiability at $x = 0$ by calculating the left-hand derivative $(Lf'(0))$ and the right-hand derivative $(R

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(D) Given $f(x) = |x|.$ We check the differentiability at $x = 0$ by calculating the left-hand derivative $(Lf'(0))$ and the right-hand derivative $(Rf'(0))$.$Rf'(0) = \lim_{h 	o 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h 	o 0^+} \frac{|h| - 0}{h} = \lim_{h 	o 0^+} \frac{h}{h} = 1.$$Lf'(0) = \lim_{h 	o 0^+} \frac{f(0 - h) - f(0)}{-h} = \lim_{h 	o 0^+} \frac{|-h| - 0}{-h} = \lim_{h 	o 0^+} \frac{h}{-h} = -1.$Since $Rf'(0) 
eq Lf'(0),$ the derivative $f'(0)$ does not exist.Therefore,the correct option is $D$.

If $f(x) = |x|,$ then $f'(0) = $

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