Sorry if I worded the question title weirdly, I'm not sure how to word it.
I used Wolfram Alpha to try to get the derivative of y^x
Here is the output (snapshot), as you can see there is a 3D plot and I get that the x and y axis are literally x and y from the equation, but what is the z-index on this plot? I would imagine it's the imaginary values, yet the plot clearly indicates this is the RE(al) part of the equation and the z-index labels have no "i" indicator in their numbers.
It looks like it's almost a scale for the x and y axis, like the values there are multiplied by the z-axis value. Having a hard time wrapping my head around why one would do that though, if that is the case it's the first time I've seen it but I don't deal with 3D Plots ever haha. Attached below:
I think complex numbers are "creeping into" this issue due to the presence of $\log$ function. In case (not shown here) that $y$ takes negative values, you may know that $\log(y)$ is defined, but as a complex entity (for example to $\log(-1)$ can be considered as $log(e^{i \pi})=i \pi$. This is why studying complex function theory is so important).