derivative of absolute value

1 ANSWERS

1. amomipais82
   

   The derivative of the real absolute value function is the signum function, sgn(x), which is defined as
   
       \sgn (x) = \frac{x}{|x|},
   
   for x ? 0. The absolute value function is not differentiable at x = 0. For applications in which a well-defined derivative may be needed, however, the subderivative is well-defined at zero. Where the absolute value function of a real number returns a value without respect to its sign, the signum function returns a number's sign without respect to its value. Therefore x = sgn(x)abs(x). The signum function is a form of the Heaviside step function used in signal processing, defined as:
   
       u(x) = \begin{cases} 0, & x < 0 \\ \frac{1}{2}, & x = 0 \\ 1, & x > 0, \end{cases}
   
   where the value of the Heaviside function at zero is conventional. So for all nonzero points on the real number line,
   
       u(x) = \frac{\sgn(x) +1}{2}.\,
   
   The absolute value function has no concavity at any point, the sign function is constant at all points. Therefore the second derivative of |x| with respect to x is zero everywhere except zero, where it is undefined.
   
   The absolute value function is also integrable. Its antiderivative is
   
       \int|x|dx=\frac{x|x|}{2}+C.
   
   Proof
   
       \int|x|dx=\int\frac{|x|}{x}xdx=\int\frac{d|x|}{dx}xdx=x|x|-\int|x|dx \iff 2\int|x|dx = x|x| \iff \int|x|dx =\frac{x|x|}{2}+C.

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