Dirac delta function is a singular distribution. So, unlike regular distributions, it isn't defined by integration against test functions because test function multiplied by Dirac delta function is undefined at $0$.
As a distribution, it is expressed as $f\{g\}=g(0)$.
\begin{align*} \int_{-\infty}^{\infty} f(x)\frac{\mathrm{d}\delta(x)}{\mathrm{d}x} \, \mathrm{d}x &= f(x)\delta(x)\biggr|_{-\infty}^{\infty} - \int_{-\infty}^{\infty} \frac{\mathrm{d}f(x)}{\mathrm{d}x}\delta(x) \, \mathrm{d}x \\ &= - \int_{-\infty}^{\infty} \frac{\mathrm{d}f(x)}{\mathrm{d}x}\delta(x) \, \mathrm{d}x \end{align*}
And I read about its property about derivative in the above formula. Here, I have doubts, how it can be derived with partial derivative regardless of its non integrable?
