For the Radon-transformation $\mathcal{R}f(r,\omega)=\int_{\{x:x\cdot\omega=r\}}f(x)\mathrm{d}\sigma(x)$ with $r\in\mathbb{R},\omega\in\mathbb{S}^{n-1}$ I want to prove the following derivative formula:
$$\mathcal{R}\lbrack\partial_if\rbrack(r,\omega)=\omega_i\partial_r\mathcal{R}f(r,\omega)$$
To do so, I did the following $$\mathcal{R}\lbrack\partial_if\rbrack(r,\omega)=\int_{\{x:x\cdot\omega=0\}}(\partial_jf)(x+r\omega)\mathrm{d}\sigma(x)$$ and $$ \omega_i\partial_r\mathcal{R}f(r,\omega)=\omega_i\partial_r\int_{\{x:x\cdot\omega=0\}}f(x+r\omega)\mathrm{d}\sigma(x)=\int_{\{x:x\cdot\omega=0\}}\omega_i\nabla f(x+r\omega)\cdot\omega\mathrm{d}\sigma(x)\\ =\int_{\{x:x\cdot\omega=0\}}(\partial_jf)(x+r\omega)\mathrm{d}\sigma(x)+\int_{\{x:x\cdot\omega=0\}}\sum_{1\leq j\leq n,j\neq i}\omega_j(\partial_jf(x+r\omega)\omega_i-\partial_if(x+r\omega)\omega_j)\mathrm{d}\sigma(x)$$
where I used that $\omega\in\mathbb{S}^{n-1}$. Unfortunately I'm not able to show that the last term vanishes. I'm thankful for every kind of help :)