At the risk of being a math nerd: Don’t say “the (second) derivative of the data”. It is the "derivative of  with respect to " (well actually you typically compute hession-vector products when using second derivatives with backprop, but hey).
Many (most?) of the loss functions are sufficiently nonlinear, but you would, of course also get second derivatives of the function specified by your neural network if you used an activation function that is curved (e.g. tanh where it is not saturated).