\int dr \frac{\partial F}{\partial f'}\delta(f') = \int dr \frac{d}{dr}(\delta f \frac{\partial F}{\partial f'}) - \int dr \delta f  \frac{d}{dr}(\frac{\partial F}{\partial f'}) = (\delta f \frac{\partial F}{\partial f'})_{r=0}^{R} - \int dr \delta f  \frac{d}{dr}(\frac{\partial F}{\partial f'})