While teaching for IIT entrance, I usually come across two types of students; One that wants to know the derivation behind every formula, and the other that just wants to use the formula to get through the exam and tests. Why to reinvent the wheel, they ask. So what is important, knowing the derivation as well or just knowing the formula with their uses in problem solving?
This is not to suggest that you should or should not learn the derivation but you should be judicious in spending time on derivation. Remember that mathematics requires lot and lot of practice and then little more practice in the limited time.
There are two types of formulas. The first type is that which has been derived by large body of complex work. We see them in simple form for our use. This gives us an impression that the formula must have been simple to derive. This may not be the case. For example, long ago, a great mathematician, Bertrand, decided to solve a question. This is famously known as Bertrand Ballot problem. There were two candidates in an election. Candidate Martin got m number of votes while candidate Nathan won n number of votes. Here m is greater than n. When the counting happens, what is the probability that Martin is always ahead of Nathan. This means at any given point, the vote count of Martin is always more than the vote count of Nathan. This was certainly not an easy task to arrive at the probability of the event. After lot of deliberations, calculations, and recursion, Bertrand arrived at the probability will be
This looks pretty simple but deriving it will take lot of time. Whether you should understand the derivation of this depends on your level of preparation. If you have completed your syllabus and have nothing better to do than go to movies, I would suggest you try to understand this derivation. However, if you have chapters left in mathematics, move on.
The second types of formulas are the ones that are derived to offer us a simple form so that we can remember it and use it for problem solving. They are not difficult to derive but the purpose of making them ready made is that they are often used in problem solving. Hence using it saves time. These are like plug and play in computer. Do we really need to know the derivation? The answer depends on whether learning the derivation adds to our understanding of the subject at hand. For example, derivation of equation of a pair of tangents drawn on a circle from a point outside the circle is a good way to understand how a line is defined. The formula is SS1 = T2. However, understanding the derivation in this case will help you in defining your approach to problem solving. For example, knowing the derivation of all the ready made equations in coordinate geometry will help you master the subject.
So use your discretion when deciding to invest time in learning the derivation. Time is precious. This is the only asset that you students have. A judicious use of time will take you ahead from your competitors.
~ Pankaj Priyadarshi