We know $|x| = \sqrt(x^2)$, determine the second derivative
$\frac{d^2}{dx^2}|x|$,
So the first derivative is sgn(x), but how do I get the second?
$\endgroup$ 23 Answers
$\begingroup$HINT:
Consider the graph of $sgn(x)$. What does it look like? What kind of slope does it have?
$\endgroup$ 9 $\begingroup$Considering derivative of discontinuity as del(x). Derivative of sgn(x) would be 2*del(x), as there exist a discontinuity at x=0 and a change in step by 2 units (from -1 to +1).
Note : This method is being used in mathematical modeling of signals. Where del(t) is an unit impluse function. And sgn is made up of two step functions.
$\endgroup$ $\begingroup$You know that
$sgn(x)=\begin{cases} 1 & x>0 \\ 0 & x=0 \\ -1 & x<0 \end{cases}$
I think you can get the derivative from there, derivate each piece of the function. Notice the discontinuity points and consider how this affects the existence of derivative at some points.
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