  Perturbation or error bounds of functions have been of great interest for a
long time. If the functions are differentiable, then the mean value theorem and
Taylor's theorem come handy for this purpose. While the former is useful in
estimating $\|f(A+X)-f(A)\|$ in terms of $\|X\|$ and requires the norms of the
first derivative of the function, the latter is useful in computing higher
order perturbation bounds and needs norms of the higher order derivatives of
the function.
  In the study of matrices, determinant is an important function. Other scalar
valued functions like eigenvalues and coefficients of characteristic polynomial
are also well studied. Another interesting function of this category is the
permanent, which is an analogue of the determinant in matrix theory. More
generally, there are operator valued functions like tensor powers,
antisymmetric tensor powers and symmetric tensor powers which have gained
importance in the past. In this article, we give a survey of the recent work on
the higher order derivatives of these functions and their norms. Using Taylor's
theorem, higher order perturbation bounds are obtained. Some of these results
are very recent and their detailed proofs will appear elsewhere.

Functional Analysis;General Mathematics