I am studying Signals and Systems.
The textbook told me $\delta'(t)$ has the following properties.
$1$. $x(t)\delta'(t-t_0)=x(t_0)\delta'(t-t_0)-x'(t_0)\delta(t-t_0)$
$2$. $\displaystyle\int_{-\infty}^t\delta'(\tau-t_0)d\tau=\delta(t-t_0)$
$3$. $\displaystyle\delta'(at+b)=\frac{1}{|a|}\delta'\left(t+\frac ba\right)$
Please prove these three equations mathematically.
Thank you.
In the theory of distributions one defines $$ D_{\delta(t-t_0)}[f]=f(t_0)\;\;,\;\;D_{\delta'(t-t_0)}[f]=-D_{\delta(t-t_0)}[f'] $$ using these definitions, we think about $x(t)\delta'(t-t_0)$ as another distribution, such that formally $$ D_{x(t)\delta'(t-t_0)}[f]=D_{\delta'(t-t_0)}[x(t)f(t)]=-x'(t_0)f(x_0)-x(t_0)f'(t_0) $$ which matches the r.h.s. of 1
For 3) one simply uses the definition $$ D_{\delta'(a t+b)}(f(t))=-D_{\delta(a t+b)}(f'(t)) $$ and then the propery of $\delta(a t+b)=1/|a| \delta(t+b/a)$. So in fact your 3. is incorrect. It should be $$\delta'(a t+b)={\rm sign}(a) \delta'(t+b/a)$$
2) is just a notation, depending on its interpretation there is nothing to prove