Differentiating a complex term by a real scalar I am struggling to understand complex differentiation. I want to find $\frac{\partial L_\theta}{\partial\theta}$ where $L_\theta=(\mathbf{x}_\theta-\mathbf{x})^\mathrm{H}\Sigma^{-1}_\theta(\mathbf{x}_\theta-\mathbf{x})$, $\mathbf{x}_\theta,\mathbf{x}\in\mathbb{C}^m, \Sigma\in\mathbb{C}^{m\times m}$ and only $\mathbf{x}_\theta$ and $\Sigma_\theta$ depend on $\theta$. To make it a little easier, $\theta\in\mathbb{R}$ is completely real. Any help would be really appreciated.
Thanks in advance.
 A: Hint:
$$\frac{\partial [(x_\theta - x)^H]}{\partial \theta} = (x_\theta -x)^{H} x_\theta 
x_\theta'$$
Observe we have just used the chain rule.
A: $\def\t{\theta}\def\p{{\partial}}\def\grad#1#2{\frac{\p #1}{\p #2}}$Let's
use a dot to denote derivatives with respect to $\t.\;$ For example
$$\dot w = \grad{w}{\t}$$
We'll also use the well-know formula for the derivative of a matrix inverse
$$\dot M^{-1} = -M^{-1}\dot M M^{-1}$$
Now your formula can be differentiate term-by-term
$$\eqalign{
\dot L_\t
 &= \dot x_\t^H\Sigma_\t^{-1}(x_\t-x)
 + (x_\t-x)^H\Sigma_\t^{-1}\dot x_\t
 + (x_\t-x)^H\dot\Sigma_\t^{-1}(x_\t-x)
\\
 &= \dot x_\t^H\Sigma_\t^{-1}(x_\t-x)
 + (x_\t-x)^H\Sigma_\t^{-1}\dot x_\t
 - (x_\t-x)^H\Sigma_\t^{-1}\dot\Sigma_\t\Sigma_\t^{-1}(x_\t-x)
\\\\
}$$

Tip: You (or more likely your professor) may think that slapping a $\t$-subscript on every variable is a great mnemonic, but it is not. It is non-standard visual clutter.
For example one normally writes
$$y = f(x) \qquad {\bf not} \qquad  y_x = f(x)$$
In most problems, the subscript positions should be reserved for component indexes, an iteration counter, or (ironically) to denote partial derivatives.
Likewise superscript positions should be reserved to denote things like conjugates, transposes, or powers. Or less often, a contravariant component index.
If you want a mnemonic, choose a good name for the variable itself; don't add a bunch of decorations like subscripts or superscripts or hats or diacritical marks.
