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I am given with Heaviside function $$H(x) = \begin{cases} 0 & if \ \ x \in (-\infty, 0) \ \\ 1 & if \ \ x \in (0, \infty) \end{cases}$$ Now 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 delta function. Now can I further find out the derivative of delta function in distributional sense?

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By definition:

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

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