Derivative with respect to complex conjugate
That's a trick which you will also find in a DSP context, that's why I choose to provide an answer here. It is related to the Wirtinger derivative, and you can find more details about it in this answer over at math.SE.

In practice this trick is often used to compute the extremum (minimum or maximum) of a real-valued function depending on a complex variable (or vector/matrix). In order to find the extremum, you formally take the derivative with respect to the complex conjugate of the variable of interest, set this derivative equal to zero, and from this equation derive the optimum value of the (possibly vector-/matrix-valued) variable.

As a simple example, take the minimization (with respect to the vector $x$) of the following error quantity:

$$\epsilon=||Ax-b||_2^2\tag{1}$$

where the vectors $x$ and $b$, and the matrix $A$ are assumed to be complex-valued. The error can be written as

$$\epsilon=x^HA^HAx-x^HA^Hb-b^HAx+b^Hb\tag{2}$$

where $^H$ denotes the Hermitian (conjugate) transpose. Now you take the derivative of $(2)$ with respect to $x^H$, formally treating this vector as being independent of $x$. The result is

$$\frac{\partial\epsilon}{\partial x^H}=A^HAx-A^Hb\tag{3}$$

Setting this derivative equal to zero and assuming that $A^HA$ is invertible, the optimal value of $x$ is given by

$$x_o=(A^HA)^{-1}A^Hb\tag{4}$$

where the matrix $(A^HA)^{-1}A^H$ on the right hand side of $(4)$ is the Moore-Penrose inverse.