I am interested in the integral of the product of a spherical Bessel function and it's derivative, for example: $$I^n_\ell(r,R)\equiv\int dq q^{n+2}j_{\ell}(qr)j_{\ell}^{''}(qR)$$ and we can keep in mind the identity $$\int dq \ q^2 j_{\ell} \left( q r \right) j_{\ell}\left( q R \right) = \frac{\pi}{2 r^2} \delta_D \left( R - r \right)$$ $I^n_\ell(r,R)$ should typically be a distribution. For example for $n=4$ we may write: $$I^4_\ell(r,R)=\frac{\partial}{\partial^2R}\int dq \ q^2 j_{\ell} \left( q r \right) j_{\ell}\left( q R \right)=\frac{\partial}{\partial^2R}\frac{\pi}{2 r^2} \delta_D \left( R - r \right)$$
However I am unsure how to evaluate the rightmost quantity. Another way of writing $I^n_\ell(r,R)$ is (for even $n$ to make things simpler): $$\int d q q^{n+2}j_{\ell}(qr)\int dr' \ q^{-2} \delta_D^{''}(r'-R)j_{\ell} \left( q r' \right)=\int dr' \delta_D^{''}(r'-R)\int d q q^{n}j_{\ell}(qr)j_{\ell} \left( q r' \right)$$
Hence I am also interested in computing integrals involving a product of Dirac delta (distribitional) derivatives, such as:
$$\int dr' \delta_D''(r'-R)f(r,r')\delta_D^{(n)}(r-r')$$
for some smooth function $f(r,r')$. How can one compute such integrals?