Derivative of delta function
You can see this by using integration by parts:

$$\begin{align}\int_{-\infty}^{\infty}x(t)\delta'(t-T)dt&amp;=x(t)\delta(t-T){\Big|}_{-\infty}^{\infty}-\int_{-\infty}^{\infty}x'(t)\delta(t-T)dt\\&amp;=x(T)\delta(t-T){\Big|}_{-\infty}^{\infty}-x'(T)\\&amp;=-x'(T)\end{align}$$

where it is assumed that $x(t)$ and $x'(t)$ are continuous at $t=T$.