If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then

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f'\left( a^+ ight) = \phi\left( a ight) f'\left( a^- ight) = - \phi\left( a ight) Here, f\left( x ight) = \left| x - a ight| \phi\left( x ight) f'\left( a^+ ight) = \lim_{h 	o 0} \frac{f\left( a + h ight) - f\left( a ight)}{h} = \lim_{h 	o 0} \frac{\left| h + a - a ight|\phi\left( a + h ight) - \left| a - a ight|\phi\left( a ight)}{h} = \lim_{h 	o 0} \frac{h \phi\left( a + h ight)}{h} = \lim_{h 	o 0} \phi\left( a + h ight) = \phi\left( a ight) Also, f'\left( a^- ight) = \lim_{h 	o 0} \frac{f\left( a - h ight) - f\left( a ight)}{h} = \lim_{h 	o 0} \frac{\left| a - h - a ight| \phi\left( a - h ight) - \left| a - a ight| \phi\left( a ight)}{h} = \lim_{h 	o 0} \frac{\left| - h ight| \phi\left( a - h ight)}{h} = \lim_{h 	o 0} - \phi\left( a - h ight) = - \phi\left( a ight)

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