  We derive iterative methods for computing the Fr\'{e}chet derivative of the
map which sends a full-rank matrix $A$ to the factor $U$ in its polar
decomposition $A=UH$, where $U$ has orthonormal columns and $H$ is Hermitian
positive definite. The methods apply to square matrices as well as rectangular
matrices having more rows than columns. Our derivation relies on a novel
identity that relates the Fr\'{e}chet derivative of the polar decomposition to
the matrix sign function $\mathrm{sign}(X) = X (X^2)^{-1/2}$ applied to a
certain block matrix $X$.
