[...] since dx/dy is the first derivative of x with respect to y, it is easy to see that these values are merely the inverse of each other. The inverse function theorem of calculus states that dx/dy = 1/(dy/dx) . The generalization of this theorem into the multivariable domain essentially provides for fraction-like behavior within the first derivative. Likewise, in preparation for integration, both sides of the equation can be multiplied by dx. Even in multivariate equations, differentials can essentially be multiplied and divided freely, as long as the manipulations are dealing with the first derivative. Even the chain rule goes along with this. Let x depend on parameter u. If one has the derivative dy/du and multiplies it by the derivative du/dx then the result will be dy/dx . This is identical to the chain rule in Lagrangian notation. It is well recognized that problems occur when if one tries to extend this technique to the second derivative [...]
But only if you factor in Janckman’s Plexus and buttonhook the inverse of the third derivative.
Then it all makes sense.