I want to differentiate the following equation to obtain an object's velocity, but I can't figure out how the eqation is actually describing the object's position.
It goes as follows: $r = (2.0 m + (3.00 m/s)t)i + (3.0 m - (2.00 m/s^2)t^2)j$, where $r = $ position at time $t$, and all quantities are in SI units.
Is $r$ a vector? And how do I differentiate or even integrate it given the complex $j$ and $i$? How do time fit in this cooridinate system?
I think that's $i$ and $j$ as in the standard unit vectors in the $x$- and $y$-directions, respectively. Not as in complex numbers. The reason I think this is because the scenario involves finding a velocity vector from a position vector, and describing the position in terms of imaginary units $i$ and $j$ is extremely outlandish, if not completely nonsensical. I think this has nothing to do with complex numbers at all.
So we should really be saying $\mathbf i$ and $\mathbf j$ to indicate they're vectors (and therefore avoid this type of confusion).
Anyway, to differentiate a vector, simply differentiate its components individually.
$$ \mathbf r(t) = (2 + 3t)\mathbf i + (3-2t^2)\mathbf j $$ Therefore $$ \mathbf r'(t) = (2+3t)' \mathbf i + (3-2t^2)' \mathbf j = 3\mathbf i - 4t \mathbf j. $$ Integrating works the same way. Integrate each component independently. And make sure to add an arbitrary constant vector (a vector with two arbitrary constant components) if your integral is indefinite, which would be unusual for vectors given my experience. Unusual but not impossible I guess.