auto-correlation of differentiator
The derivative of the Dirac delta impulse is written as $\delta'(\tau)$. This helps with notation because the mistake you made is to write $h(-\tau)=\frac{d}{d\tau}\delta(-\tau)$, which is not the case because $\delta(\tau)$ is an even (generalized) function, whereas the derivative operator $\delta'(\tau)$ is an odd (generalized) function:
$$\delta'(\tau)=-\delta'(-\tau)\tag{1}$$
Consequently, the auto-correlation function of the output of the differentiator is
$$\begin{align}R_{YY}(\tau)&amp;=R_{XX}(\tau)\star\delta'(\tau)\star\delta'(-\tau)\\&amp;=-R_{XX}(\tau)\star\delta'(\tau)\star\delta'(\tau)\\&amp;=-\frac{d^2}{d\tau^2}R_{XX}(\tau)\tag{2}\end{align}$$