Implicit Differentiation and Recursion

This weekend, I had to work on my Calculus (MAT137Y1Y) Problem Set #4, which is due this coming Tuesday. One of the questions involved an equation defined implicitly in terms of x and y, and asked us to differentiate it in two different ways: 1) by making the implicit definition an explicit equation of y = f(x), and then differentiating, and 2) by implicitly differentiating the given equation.

Since the function was already given implicitly, it was messy and tedious to turn it into an explicit definition. It took more time, more care, more calculations, and resulted in the same answer as the implicit differentiation process. On the other hand, implicitly differentiating it was quick, simple, and (most importantly) correct. I guess the professor was trying to prove to us that, although implicit functions can be made explicit and then derived (but perhaps not always), problems expressed implicitly are efficiently handled with implicit differentiation.

Sound familiar?

Dealing with an implicit definition in an explicit way is similar to an iterable approach to a recursive coding problem, whereas implicit differentiation is much like recursion. Sometimes, although the result is the same, approaching a problem with iterations will take more time and will be much messier, if it's even possible to approach the problem this way. Recursion goes straight to the core of the problem and gives the correct result much more elegantly.

Much like approaching the implicit definition explicitly, this week's lab ended with a challenge to iterably program the count_less method, which was easily programmed with recursion. Recursion and implicit differentiation are extremely useful and powerful in certain situations, as my problem set and Computer Science lab taught me, so to my professors I say: lesson learned.