how to prove that the derivative of this expression(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

f(x)=-\frac12x^2,x<0

[/tex]

[tex]

f(x)=\frac12x^2,x>=0

[/tex]

is f'(x)=|x|

i tried

[tex]

\lim _{x->0^-}\frac{f(x)-f(0)}{x}=\lim _{x->0^-}\frac{-\frac12x^2-0}{x}=\lim _{x->0^-}-\frac12x=0\\

[/tex]

[tex]

\lim _{x->0^-}\frac{f(x)-f(0)}{x}=\lim _{x->0^-}\frac{+\frac12x^2-0}{x}=\lim _{x->0^+}+\frac12x=0

[/tex]

but i get values

it doesnt show that f'(x)=|x|

??

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# How to prove that the derivative f this function is |x|

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