Derivatives and rates of the stretch and rotation tensors

General expressions for the derivatives and rates of the stretch and rotati on tensors with respect to the deformation gradient are derived. They are b oth specialized to some of the formulas already available in the literature and used to derive some new ones, in three and two dimensions. Essential i ngredients of the treatment are basic elements of differential calculus for tensor valued functions of tensors and recently derived results on the sol ution of the tensor equation AX + XA = H in the unknown X.