Mathematicians, engineers and a host of other professionals ranging from economists, earth-scientists to astronomers need to know how to apply mathematical formulae in their everyday, professional lives. The importance of mathematical formulae cannot therefore be over-emphasized. We learn about many mathematical formulae through our school years, the number and variety of mathematical formulae we encounter in our learning life depending on the level of mathematics we acquire.
There are very many formulae to remember, especially if one is preparing to sit one of the many academic exams in one's studying career. And although thankfully, many school and examination boards have begun the practice of providing a list of essential formulae to candidates taking mathematics exams, there are still many important mathematical formulae to memorize, failure to be able to memorize or derive these formulae may be fatal in that it will result in the candidate's inability to solve the problems set by the examiners.
Taking a very simple example, that of the surface area of a cylinder of base radius r and height h, we recall that the formula is
S = 2 x pi xr x h, where 'pi' is the number 22/7, r is the base radius of the cylinder and h the height of the cylinder. If we picture in our mind a cylinder sitting on a flat table, we could visualize its shape, its height and its circular base, of radius r. Now if we put a piece of thin paper and wrap around the surface of the cylinder so that the paper completely covers the surface, and not more, then when we take the paper off the cylinder it should assume the shape of a rectangle of length (which corresponds to the circumference of the base circle) 2x pi x r, while the breadth of the rectangle would be h, the height of the cylinder. It would therefore be obvious that, since the area of the rectangle is length x breadth, the surface area of the cylinder (which is the area of the paper rectangle) is derived as the product of the three terms, namely, 2xpi, r and h.
Taking a much more advanced example of mathematical formulae, we find in integral calculus the useful formula for the integration of a product of two functions such as simple algebraic function as x and a trigonometric function as sin x. The formula we need to apply to integrate the product (x sin x) with respect to x requires the technique known as "integration by parts". From my experience as a mathematics teacher, many students (even those taking the UK Advanced Level Math ) could not remember this formula. Fortunately for them, this formula is in fact provided in the Formulae List when students take the relevant exam. But my point is that it is not difficult to derive this formula if the student remembers the rudiments of differential calculus. If the student remembers the formula for differentiation of a product of two functions, then it is a matter of applying the fact that integration is the "opposite" or "reverse" process of differentiation in order to derive at the relevant formula for such an integration.
To summarize, the general principle of deriving a mathematical formula is to gather the basic elements of an issue or problem (represented in algebraic forms, in symbols such as x, y, z, S, A etc) and use our ability to think logically and weave a coherent picture of the problem at hand and then use 'trial and error' for a start. Of course some brilliant mathematicians form and derive their formulae straight in their brain without resorting to experimentation. But generally, faced with a new and novel situation, even the best engineer or scientist has to "improvise" and experiment with a 'prototype' formula and improve on it by putting the preliminary formula to practical tests.
For instance if an engineer needs to calculate the cost of building a jetty, he or she needs to gather all the relevant data - such as the design of the jetty, materials to be used, the amount of the said materials, man-power etc etc and then he sets out to derive a mathematical formula, using the data and the knowledge he has acquired as an engineer. He has to 'derive' his own formula, although there are standard formulae to apply for computing quantities such as maximum force that the jetty could withstand, the force of the waves that will hit the jetty, etc. And if his formula has been correctly derived, he will be able to compute the costs of building fairly accurately; otherwise, he has to improve his formula by taking into account factors which are material and which he might have ignored as immaterial in his earlier derivation of his formula.