This is a simple and maybe stupid question. Can you take a derivative of a vector component with respect to a vector? Or even more generally,can you take the derivative of a component of a tensor with respect to the whole tensor? For instance in the cauchy tensor could you take the xx component of the tensor and compare it to the rest of the components of the tensor by taking the derivative of xx component with respect to the rest of the tensor. I know you can take deriviatives of tensors, but can you take derivatives of components treating the rest of the components like dependent variables?