Distributional Derivative of delta function.

Question

I am given with Heaviside function $$ H(x) := \begin{cases} 0, & \text{if } x \in (-\infty, 0) \\ 1, & \text{if } x \in (0,\infty). \end{cases} $$ I have calculated its distributional derivative $$T_{H'}(x) = -\int_{-\infty}^{\infty} H(x) \phi'(x) dx$$ where $\phi$ is a test function and $T_{H'}$ represents distributional derivative. After calculating it I get $$T_{H'} = \delta,$$ where $\delta (\phi(x)) = \phi(0)$ is the delta function. How can I find out the distributional derivative of the delta function?

Answer 1

By definition:

$$\delta'(\phi) = -\delta (\phi') = - \phi'(0)$$