  The norm of the $m$th derivative of the map that takes an operator to its
$k$th antisymmetric tensor power is evaluated. The case $m=1$ has been studied
earlier by Bhatia and Friedland [R. Bhatia and S. Friedland, Variation of
Grassman powers and spectra, Linear Algebra and its Applications, 40:1--18,
1981]. For this purpose a multilinear version of a theorem of Russo and Dye is
proved: it is shown that a positive $m$-linear map between $C^{\ast}$-algebras
attains its norm at the $m$-tuple $(I, \, I, ..., I).$ Expressions for
derivatives of the maps that take an operator to its $k$th tensor power and
$k$th symmetric tensor power are also obtained. The norms of these derivatives
are computed. Derivatives of the map taking a matrix to its permanent are also
evaluated.
