We derive methods to compute higher order differentials (Hessians and Hessian-vector products) of the rendering operator. Our approach is based on importance sampling of a convolution that represents the differentials of rendering parameters and shows to be applicable to both rasterization and path tracing. We demonstrate that this information improves convergence when used in higher-order optimizers such as Newton or Conjugate Gradient relative to a gradient descent baseline in several inverse rendering tasks.