Distributional Derivative of a Piecewise function

Question

Use the definition of the distributional derivative to find the first-order derivative (in the distributional sense) of the function \begin{align*} h(x) = \left\{ \begin{array}{lllll} 0, & \quad x < -2\\ 2, & \quad -2 \leq x \leq -1\\ 0, & \quad -1 < x < 1\\ 1, & \quad 1 \leq x \leq 2\\ 0, & \quad x > 2 \end{array} \right. \end{align*}

I assume here that i would have to split the integral into multiple parts and then solve it, but not sure how to proceed.

Any tips would be appreciated!

Answer 1

$\int h \phi '=2\phi (-1)-2\phi (-2)+\phi (2)-\phi (1)$ and hence $h'=-2\delta_{-1}+2\delta_{-2}-\delta_2+\delta_1$.