Observe that the above expression is just the limit of the difference quotients for . It is the same limit that we studied as the solution to the tangent problem and whose value we called . Thus far, we have encountered two apparently different situations in which the same mathematical object, namely, the derivative, arises. This should make us suspect that we are dealing with something truly wonderful, a tool that can be used in many different scientific and other applications. And of course, history has shown that our suspicions are correct.
Derivatives are worthy of study in their own right, as tools waiting to be applied.
Following such a program, we will be developing rules for calculating derivatives. These rules will come in handy when the derivative has a physical or scientific meaning such as velocity or acceleration. The power of calculus is that many of the computational rules are algebraic. For now, we will be calculating derivatives from the limit definition and accumulating rules as we go along. See if you can discover some of the rules for yourself.