>>14425215>how so? explainYou mentioned "slope of the tangent" which is a thing we might care about if we were looking at something on the xy plane, where we usually place function inputs and outputs. The line is created by looking at the position on the y axis as a function of the position on the x axis. Then if you draw a tangent line you can see that its slope is the same as the slope at the point it is tangent to, and particularly useful, its slope is the ratio that y changes with respect to a change in x, that is, the derivative of y with respect to x.
Now complex numbers, as other anons have rightly pointed out, are some kinda clunky vector space fused with a special redefinition of multiplication such that it means rotation instead of scaling all for the special purpose of getting a number that can reasonably be said to equal the square root of a negative number.
So the complex plane we are not looking at function input and output like we are on the xy plane, we are looking at the """imaginary""" aka rotating, and """real""" aka scaling components of a single complex number.
Nobody calls them rotating and scaling components, but imo it would be much better than the stupid names we got stuck with due to stupid historical reasons.
Anyway. You have a function of x though, and it's making a shape on the complex plane, but there is no x axis on the complex plane so you are not seeing the "change in x" so no tangent you draw here is going to have any relationship to your derivative with respect to x.
If you wanted to add an x axis, which seems reasonable, you would need to have it extending out perpendicular to the complex plane. Perhaps imagine a spring or screw rising out of it showing how for every value of x there is exactly one point on the complex plane which is output by the function, and the rate that that point is moving for every nudge to x has both a magnitude, and a direction, or said another way, it has a real and imaginary component.
