We start with our quantity theory of money, which tells us what nominal income PY is:
PY = MV(i)
Normally one would stabilize nominal income by boosting M. The problem is that the normal way the central bank boosts M is via open-market operations by which it buys safe savings vehicles for cash. Reduce the supply of safe savings vehicles and you raise their price--and thus lower the safe nominal interest rate i some more. Lowering i further reduces the opportunity cost of holding money and further reduces V. The net effect on PY is:
d(PY) = [V(i) + M(dV/di)(di/dM)]dM
Now, this all seems pretty scientific. Its logic is best exemplified by Don Patinkin’s Money, Interest and Prices (1957), which had page after page of such equations, and has been called the 'height of the IS/LM' theory. It appeared to be a very compelling theory, because it was consistent and logical at every step. All the partial derivatives made sense, but one had to hope that such simple models were sufficient to overcome all the variables and interactions not explicitly addressed, because in a complex economy there's an almost infinite number of relationships and omitted factors.
But what are theses variables and relationships that have such mathematical precision? Consider V, velocity. Velocity is the residual of the measurables nominal income and money. Thus, the derivative of V with respect to i (the interest rate) is really from the derivatives, and their correlations, of nominal income and money. Are either of these stable in any sense (d(PY)/di, dM/di)? No. They have no stable values and suggest they are no better than asserting a mathematical relationship between your body temperature and how much coffee you drank based on thermodynamics: there's a simple effect from the initial impact, but very shortly feedback effects that make the initial physical model worthless.
And so it goes with all these relationships. The economy is a complex, nonlinear, adaptive system where short run effects are often opposite of long run effects. Ultimately, people need to be involved in activities that generate more benefit than they they cost for they to be sustainable, and they must appear attractive to individuals given their other opportunities as they see them. A higher growth rate simply involves having more people near their unknown optimal--unknown because only rudimentary laborers know what the truly best use of their time is at any moment--and stagnation is caused when too many individuals lose their ability or willingness to actively search and make these choices. The more government puts in rules saying we can't drill in the Arctic because of potential effects of exhaust on a nearby village of 100 Eskimos, or that we can't cut hair without a state license, the lower our productivity. Economic freedom results in long term growth and so the key to macro policy is not from some chain rule but rather the wisdom of Smith, Bastiat, von Mises, Hayek, Friedman, Stigler, and the rest who found aggregate prosperity and economic freedom synonymous.
Increasing the money supply to fool employees into thinking they have more money than they do, hoping on a multiplier effect via such derivative compounding logic, is simply absurd. Currently, we are changing M to affect i, so the relationship should be di/dM, not dM/di, highlighting why the empirical relationship between i and M is so spastic. If countries could simply print and spend to increase GDP, the 1970's would have been an era of great growth, instead of the disaster it was, which is why the 1970s--increasing inflation, increased government spending--is such a profound empirical contradiction to the Keynesian model.
Thinking about aggregates this way is pure blather, like the way Marxist intellectuals talked about the laws of motion a century ago when their Oracle bloviated in Das Kapital that just 'as the heavenly bodies, once thrown into a certain definite motion, always repeat this, so it is with social production.' Yeah, just like celestial mechanics. I generally just ignore any argument based on or alluding to such pretentious, hopeful, willfully naive twaddle, but if for some reason someone accosted me at a Town Hall meeting and said such an equation implied we should do X, I would simply ask that they show me the regression results for it on a handful of developed countries in the modern era. The parameters would not be consistent over times or between countries.