Not neccessarily.
The reciprocal x crop factor rule is just a starting point and as such would vary with the focal length at which you are shooting. However, since it is so easy to increase the focal length (zoom in) with a zoom lens, it might be safer to use a shutter speed which is the reciprocal of the longest focal length x crop factor.
This rule of thumb has many-many variables, including your physical condition (both health and tempory condition from running, etc,), your technical ability to hold a camera steady (breathing, gentle shutter release, etc.), your environment (wind, etc.), the weight and size of your lens and camera and many more. It can also be impacted by the adrenaline running through your body during exciting moments and the amount of caffine, nicotine or alcohol you consume.
I have found that this rule holds pretty true for me in the shorter focal lengths (100mm and under) but, when using long focal length lenses, I need to increase the shutter speed beyond the reciprocal of focal length x crop factor. That is why I really like IS equipped lenses in long focal lengths.
As an example, I can hand-hold my 70-200mm f/4L IS lens at 1/60 second and be assured of almost 100% sharp imagery when the IS is turned on and I am at 200mm. There is no way I could come close to this without IS. In fact, because of IS, I use the 70-200mm f/4L IS lens 3-4x more often than I ever used the non-IS version of that lens.