The Coriolis Effect mysteriously appears with the change from inertial to rotational reference frames, but I believe it can be best understood through thought experiment rather than a partial derivative derivation since the mathematics as a tool comes after physical understanding.
When you go towards the poles, you speed up in the direction of the spinning of the Earth since your angular velocity must increase if you decrease your radius from the axis of rotation without changing your angular momentum. If you start at a pole and go towards the equator, you must decrease your angular velocity since your radial distance from the axis of rotation is increasing and your angular momentum is the same; you will appear to be deflected West while actually the Earth’s angular velocity stays the same and yours is relatively slower.
Thus in the Northern Hemisphere, you have Westerlies from the North and Easterlies from the South. The Westerlies and Easterlies can intermingle, causing a cyclone when a Westerly hits the West side of an Easterly and starts to rotate anticlockwise. An anticyclone is formed when an Easterly rams into a Westerly on the West of the Westerly.
Two specific scenarios can be calculated with no knowledge of the cross-product. When you are traveling around the outside of a rotating circle and when you are traveling along the radius of a rotating circle. In fact, the latter situation will get you magnitude close enough for use in meteorology; you merely have to decide which direction to apply these based on the conservation of angular momentum we already talked about.
Since angular momentum is equal to mass times tangential velocity times radius, angular momentum is also equal to mass times angular velocity times radius times radius, the second radius canceling with the implicit radius in the definition of angular velocity as tangential velocity divided by radius. If we are changing radius with time, we can differentiate radius with respect to time; angular velocity and mass are held constant. Since the derivative of momentum with respect to time is force, thus the angular force by which nature keeps the angular momentum constant is 2mωr dr/dt. Change in distance over time is velocity so 2mωrv is the torque. To convert from a torque to a force we divide by r: 2mωv In the case of meteorology, r is equal to the radius of the earth times the sine of the latitude and ω is equal to the angular velocity of the Earth. With this formula for magnitude and knowledge of which direction the force ought to act, East or West, as we discussed in paragraph one, you can determine the resulting meteorologically relevant Coriolis effect.
However, this does not completely explain the Coriolis force. If you are going West or if you are going East, you will still experience a deviation. Imagine you are looking at the Earth from above and watching a storm race along the circumference as the Earth, which looks like a disk, turns. You see the centripetal acceleration as -mv²/r, but the storm v² is not merely the speed of the turning planet, but rather the sum of the turning planet and the storm’s additional speed; -m(v-storm + v-earth)²/r. Distributing out terms, -m((v-storm)² + v-storm(v-earth) + v-earth(v-storm) + (v-earth)²)/r is obtained. Reorganizing, we get, -m(v-storm)²/r - 2m(v-storm)(v-earth)/r - m(v-earth)²/r. Since ω = v-earth/r, we can restate this as -mv²/r - 2mvω - mωr. This equals the centripetal force. But the storm, knowing only that it is running along the circle, not knowing that it is in a rotating system as well, sees only the first terms as the centripetal acceleration. The other two forces are mysterious presences. - 2mvω is the Coriolis “force” and - mωr the centrifugal “force”.* In this case, the Coriolis force is different from before.
The question of interpolating between these two magnitudes is different from the question of understanding the Coriolis effect because it can only be taught through mechanistic calculation with cross-products. Cross products are insulting to the intuition because they represent a rotation with a direction; going around one way is represented as an arrow going up, going around the other way is represented as an arrow down. These objections have been remedied in the language of differential forms where torques are oriented areas, but this language is often overlooked by engineers, and, unfortunately, it is best to use a confusing standard than to confuse the standard. So I will give you a conventional formula and we will both try to go home happy; perhaps with an interest in returning to reformulate later: -2m(ωxv). Note that since ω is “up,” the cross product with a velocity going around the circumference like the storm was will be sin(π/2) = 1 and will go inward, and, since for the changing radius case where the radius is going straight out into space with respect to time, sin(π/2) = 1 and the force will go West, our derivations hold as specific cases of this more general formula.
*On the Earth, the centrifugal force has gone away, over time, thanks to the centrifugal force pulling the Earth into an oblate spheroid such that gravity now pulls in the precise manner that takes away this force.
See Also: James Price's Coriolis Tutorial