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Question Number 74959 by aliesam last updated on 04/Dec/19

$$f\left(x\right)=\mid x+5\mid -\mid x-2\mid -\mid x+6\mid$$$$$$$$find{f}^{\prime }\left(x\right)$$

Commented by mathmax by abdo last updated on 04/Dec/19

$$x-\mathrm{\infty }-6-52+\mathrm{\infty }$$$$\mid x+5\mid -x-5-x-5x+5x+5$$$$\mid x-2\mid -x+2-x+2-x+2x-2$$$$\mid x-6\mid -x+6x-6x-6x-6$$$$f\left(x\right)x-13-x-1x+9-x+13$$$$sox<-6\Rightarrow f\left(x\right)=x-13\Rightarrow f^{{}^{\prime }}\left(x\right)=1$$$$-6<x<2\Rightarrow f\left(x\right)=-x-1\Rightarrow f^{{}^{\prime }}\left(x\right)=-1$$$$-5<x<2\Rightarrow f\left(x\right)=x+9\Rightarrow f^{{}^{\prime }}\left(x\right)=1$$$$x>2\Rightarrow f\left(x\right)=-x+13\Rightarrow f^{{}^{\prime }}\left(x\right)=-1$$

Answered by mind is power last updated on 04/Dec/19

$$\mathrm{g}\left(\mathrm{x}\right)=\mid \mathrm{x}\mid$$$$\mathrm{\forall }\mathrm{x}\in \mathbb{R}-\left\{0\right\}$$$${\mathrm{g}}^{\prime }\left(\mathrm{x}\right)=\mathrm{sign}\left(\mathrm{x}\right)$$$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)=\mathrm{sign}(\mathrm{x}+5)-\mathrm{sign}(\mathrm{x}-2)-\mathrm{sign}(\mathrm{x}+6)$$$$\mathrm{\forall }\mathrm{x}\in \mathbb{R}-\{-6,-5,2\}$$

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