$$ \delta^3(\vec{\mathbf{r}}-\vec{\mathbf{r}}_0)=\delta(x-x_0)\delta(y-y_0)\delta(z-z_0) $$
$$ \int_{\textit{all space}}\delta^3(\vec{\mathbf{r}}-\vec{\mathbf{r}}_0)d\tau=1 $$
$$ \delta(x-\alpha)=\frac1{2\pi}\int_{-\infty}^\infty e^{ip(x-\alpha)}dp. $$
$$ \delta^3(\vec{\mathbf{r}}-\vec{\mathbf{r}}0)=\frac 1{(2\pi)^3}\int{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty e^{ip(x-x_0)}e^{ip(y-y_0)}e^{ip(z-z_0)}d^3p. $$
$$ \int_{-\infty}^\infty\delta(\alpha x)\operatorname{d}x=\int_{-\infty}^\infty\delta(u)\operatorname{\frac{du}{|\alpha|}}=\frac1{|\alpha|} $$
The derivative of the delta distribution, denoted $\delta'$ and also called the Dirac delta prime or Dirac delta derivative as described in Laplacian of the indicator.
$$ \delta'[\varphi]=-\delta[\varphi']=-\varphi'(0). $$