Three reasons come to mind.Originally Posted by lingyai
The first is that your function may not be solvable. In that case numerical approximation needs to be done and the derivative method, though slightly incorrect, may be much faster than the numerical solution.
The second is that you may simply have zillions of variables to consider. Again you might need to resort to numerical calculation with the above comments applying again.
Finally, a note about Excel's (and many other program's) techniques for solving equations...a number of equation solvers use Calculus in solving equations...almost with certainty if you are doing numerical approximations the program is going to be using Calculus. So theoretically you are using Calculus anyway.
Basically my point is that you are right as you refer to the type of problem you are looking at. But in solving a general problem you might need to resort to other, less "black-box" like, techniques. And hey, the derivative method gives a simply solved first approximation, so why not use it?