Thousands of you have begged me to strip the camouflage from "Differentials for Dummies" to isolate the fallacy. Very well: here is as minimal a version as I could come up with.
Theorem: Any twice differentiable function of one variable is linear.
y''/y'2 = (d2y/dx2)/(dy/dx)2 = (d2y/dx2) (dx/dy)2
= (d2y/dx2) (dx2/dy2) = d2y/dy2 = 0
Therefore y'' = d2y/dx2 = 0.
Therefore y = Mx + N.
Note M cannot = 0, else y''/y'2 is indeterminate; however, constant functions are also linear.
It's a bottle of wine.