I am still in high school and am a part of the robotics club that competes in the FTC (First Tech Challenge). I am just about finishing my first Calculus class (Calc 1), and would be ecstatic to be able to apply this someway in a real world example such as robotics. [Besides PID. It seems like only approximations anyways]
So far, I've only been working with "fabricated" math problems. Would deriving an equation from real life situations be too complicated?
It all depends on how good you want your system to be and how much effort you're willing to put in. An over-designed robot with good performance can generally be done evaluating the robot on a per-joint basis and usually applying PID control at each joint.
A highly optimized robot with stellar performance will be designed as a system. Here you have to look at:
kinematics - if this joint turns 20 degrees, how far does that thing move?,
physics - what is the moment of inertia and equivalent point mass for that thing?,
dynamics - knowing moment of inertia and how far that thing is moving, what's its relationship between power and acceleration?
datsheets - is this joint's actuator capable of delivering enough physical power for my desired accelerations? What control signal to this joint achieves a given joint acceleration?
control theory - knowing now how this joint's rotation affects that thing and what control signal gives me a desired joint acceleration, what should my desired acceleration be to get that thing to do what I want?
Note that control theory may involve another branch called inverse kinematics, and while basic control theory may involve "just" PID, if you have anything more complicated than a single input single output (SISO) system you will probably wind up doing some more complex control like state-space, which is also capable of handling inter-actuator effects.
Physics and control theory are where you will use most of your calculus, but it is useful to know it for everything else as well so you can see why and when one formula is applicable when another is not. This is usually due to simplifying assumptions used when solving the base differential equation.