Memory is a limiting resource for many deep learning tasks.Beside the neural network weights, one main memory consumer is the computation graph built up by automatic differentiation (AD) for backpropagation.We observe that PyTorch's current AD implementation neglects information about parameter differentiability when storing the computation graph.This information is useful though to reduce memory whenever gradients are requested for a parameter subset, as is the case in many modern fine-tuning tasks.Specifically, inputs to layers that act linearly in their parameters and inputs (dense, convolution, or normalization layers in evaluation mode) can be discarded whenever the parameters are marked as non-differentiable. We provide a drop-in, differentiability-agnostic implementation of such layers and demonstrate its ability to reduce memory without affecting run time on popular convolution- and attention-based architectures.