  We present theory for general partial derivatives of matrix functions on the
form $f(A(x))$ where $A(x)$ is a matrix path of several variables
($x=(x_1,\dots,x_j)$). Building on results by Mathias [SIAM J. Matrix Anal.
Appl., 17 (1996), pp. 610-620] for the first order derivative, we develop a
block upper triangular form for higher order partial derivatives. This block
form is used to derive conditions for existence and a generalized
Dalecki\u{i}-Kre\u{i}n formula for higher order derivatives. We show that
certain specializations of this formula lead to classical formulas of quantum
perturbation theory. We show how our results are related to earlier results for
higher order Fr\'echet derivatives. Block forms of complex step approximations
are introduced and we show how those are related to evaluation of derivatives
through the upper triangular form. These relations are illustrated with
numerical examples.
