First let's try to understand why the derivative of the function $f$ given by $f(x) = x^2$ is equal to $2x$ and not to $x$. (The product rule and the power rule are both generalizations of this.)
Imagine that you have a square whose sides have length $x$. Now imagine what happens to its area if we increase the length of each side by a small amount $\Delta x$. We can do this by adding three regions to the picture: two thin rectangles measuring $x$ by $\Delta x$ (say one on the right of the square and another on the top) and one small square measuring $\Delta x$ by $\Delta x$ (say added in the top right corner.) So the change in the area $x^2$ is equal to $2x \cdot \Delta x + (\Delta x)^2$. If we divide this by $\Delta x$ and take the limit as $\Delta x$ approaches zero, we get $2x$.
So geometrically what is happening is that the small square in the corner is too small to matter, but you have to count both rectangles. If you only count one of them, you will get the answer $x$; however, this only tells you what happens when you lengthen, say, the horizontal sides and not the vertical sides of your square to get a rectangle. This is a different problem than the one under consideration, which asks (after we put it in geometrical terms) how the area varies as we lengthen all the sides.
You are confusing things. If I define $f : \mathbb{R} \to \mathbb{R}$ by $f(x)=x^n$ this is very different from defining $g: \mathbb{R} \to \mathbb{R}$ by $g(x)= a^x$, note that in the first one the exponent is not varying and on the other function the variable appears on the exponent.
For the first function the derivative is just $f'(x) = nx^{n-1}$, for the second one things are different. First it turns out that first you need to define what it means to raise something to a real number (notice that the usual definition doesn't work, what would mean multiplying a number by itself $\pi$ times?), in that case for reasons that I won't explain here we define this function as:
$$a^x = e^{x\ln a}$$
In that case, if we know how to differentiate $e^x$ (and usually when we construct this, we already know), we'll have the following:
$$(\ln \circ g)(x)=x \ln a$$
Now the chain rule gives:
$$\ln'(g(x))g'(x)=\ln a$$
However $\ln'(x) = 1/x$ because of the construction of $\ln$ and $g(x)=a^x$ so tha we have:
$$\frac{1}{a^x}g'(x)=\ln a \Longrightarrow g'(x) = a^x \ln a$$
Notice that there was a crucial appeal to the definition $a^x = e^{x \ln a}$. To know why we define things this way look at Spivak's Calculus, there's an entire chapter devoted to all the constructions about logs and exponentials.
Best Answer
The values taken by the function $f : x \in \mathbb{R} \, \longmapsto \, ix$ are complex numbers. However that's not a big deal. You can still define the derivative of $f$ using a limit. Given $x \in \mathbb{R}$,
$$ f'(x) = \lim \limits_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim \limits_{h \to 0} \frac{i(x+h) - ix}{h} = i. $$