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While calculating an inverse Laplace transform Wolfram Alpha returned to me the following output:

7 + 2 DiracDelta[-1 + t] + 14 DiracDelta[t] + HeavisideTheta[-1 + t] + 16 DiracDelta'[t]

What does DiracDelta'[t] mean? A derivative of Dirac Delta function? Wouldn't that be infinite at $0$ and zero everywhere else? That is, basically the Dirac Delta function itself?

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"Infinite at zero and zero everywhere else" is a woefully inadequate description of the dirac delta.

The best (and usually literal) definition of the dirac delta is basically that the notation resembling an integral containing a dirac delta is defined to mean evaluation:

$$ \int_{-\infty}^{\infty} f(x) \delta(x-a) \, \mathrm{d}x := f(a) $$

whenever $f$ is continuous at $a$. Notation involving the derivative is defined by a similar formula:

$$ \int_{-\infty}^{\infty} f(x) \delta'(x-a) \, \mathrm{d}x := -f'(a) $$

where $f$ is continuously differentiable at $a$. The idea behind the definition is that it is is meant to invoke partial integration; to imagine a hypothetical calculation

$$ \int_{-\infty}^{\infty} \left( f(x) \delta'(x-a) + f'(x) \delta(x-a) \right) \, \mathrm{d}x = (f(x) \delta(x-a))\big|_{x=-\infty}^{x=\infty} = 0 $$

There is a systematic approach to this sort of stuff: they're called distributions. On a suitable space of test functions, this partial integration formula is the definition of the derivative.

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