The simple answer is that the 'usual facts' about derivatives for real functions do not automatically transfer to complex functions. For example, $\ln(|z|)$ does not have a complex derivative with respect to $z$. So your question is a good one; since you are not assuming that properties for reals carry over to complex numbers. (See this post for another common kind of error due to such assumptions.)
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\def\lfrac#1#2{{\large\frac{#1}{#2}}}
$
So what holds? If you build a more general framework for differentiation, such as sketched in this post, you can establish many basic properties of derivatives, of which I shall list some below. This framework is not only more intuitive, it also clearly shows why complex differentiability is much stronger than real differentiability.
Specifically, given any real variables $x,y$ varying with real parameter $t$, for $\lfrac{dx}{dy}$ to be defined when $t = p$ we merely need $\lfrac{Δx}{Δy}$ to have a limit as $t \to p$. This is along a path. In contrast, given complex variables $x,y$ varying with complex parameter $t$, for $\lfrac{dx}{dy}$ to be defined when $t = p$ we need $\lfrac{Δx}{Δy}$ to have a limit as $t \to p$ along all possible paths.
But since the definition itself does not care what type the parameter is, nor what type the variables are, except that the values come from a field with a norm, all the basic properties that arise from algebraic identities hold with the same proof. Time for the examples.
Given any complex variables $x,y,z$ varying with some parameter $t$:
$\lfrac{dc}{dx} = 0$ whenever $\lfrac{dc}{dx}$ is defined, for any constant $c$.
$\lfrac{d(c·y)}{dx} = c·\lfrac{dy}{dx}$ whenever $\lfrac{dy}{dx}$ is defined, for any constant $c$.
$\lfrac{d(y+z)}{dx} = \lfrac{dy}{dx}+\lfrac{dz}{dx}$ whenever $\lfrac{dy}{dx},\lfrac{dz}{dx}$ are both defined.
$\lfrac{dx}{dz} = \lfrac{dx}{dy}·\lfrac{dy}{dz}$ whenever $\lfrac{dx}{dy},\lfrac{dy}{dz}$ are both defined.
$\lfrac{d(y·z)}{dx} = \lfrac{dy}{dx}·z+y·\lfrac{dz}{dx}$ whenever $\lfrac{dy}{dx},\lfrac{dz}{dx}$ are both defined.
$\lfrac{d(x^n)}{dx} = n·x^{n-1}$ whenever $\lfrac{dx}{dt}$ is defined, for any natural $n \ge 1$.
$\lfrac{d(\exp(x))}{dx} = \exp(x)$ whenever $\lfrac{dx}{dt}$ is defined.
$\lfrac{d(\cos(x))}{dx} = -\sin(x)$ whenever $\lfrac{dx}{dt}$ is defined.
$\lfrac{d(\sin(x))}{dx} = \cos(x)$ whenever $\lfrac{dx}{dt}$ is defined.
$\lfrac{d(\ln(x))}{dx} = \lfrac1x$ whenever $x \notin \mathbb{R}_{\le 0}$ and $\lfrac{dx}{dt}$ is defined. (Here taking principal branch for $\ln$.)
In particular, we can see how these apply to your examples:
$\lfrac{d(\sin(3it))}{dt} = \lfrac{d(\sin(3it))}{d(3it)}·\lfrac{d(3it)}{dt} = \cos(3it)·3i$. Here $t$ is a real parameter.
$\lfrac{d(\sin(3iz))}{dz} = \cos(3iz)·3i$ where $z$ is a complex parameter too, corresponding to the fact that $( z \mapsto \sin(3iz) )$ has a complex derivative.
$\int \exp(i2t) dt = \lfrac1{2i}·\exp(i2t) + c$ for some constant $c$, because $\lfrac{d(\lfrac1{2i}·\exp(i2t))}{dt} = \exp(i2t)$ everywhere.
If $z = x+y·i$ where $x,y$ are real variables varying with real parameter $t$, then by the above properties $\lfrac{dz}{dt} = \lfrac{dx}{dt}+\lfrac{dy}{dt}·i$. Even if the expression for $x$ uses complex numbers, you can use the above properties to compute $\lfrac{dx}{dt}$. For example if $x = \exp(it)+\exp(-it)$, which is always real, then $\lfrac{dx}{dt} = i\exp(it)-i\exp(-it)$.
