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Now that we have an idea of what the derivative is at a conceptual and geometric level, it’s time to dig into the actual algebra! As we now know, the derivative is nothing more than the classic difference quotient we used when calculating average rates of change, combined with the mechanic of evaluating limits to determine an instantaneous rate of change. We’ve covered this in detail when we discussed instantaneous rates of change, so this should hopefully all seem at least a little bit familiar.

So, what does it mean to try to calculate the derivative of some function $f(x)$ at an $x$-value $b$? This is equivalent to trying to determine the instantaneous rate of change of $f(x)$ at $b$, which means we want to solve the following difference quotient: $$\lim \limits _{x\rightarrow b} \frac {f(x) - f(b)}{x-b}$$

Often it is easier to calculate this slightly limit in a slightly different form though. Instead of thinking about the second point merging to the first, we will instead track how far away the second point is as it merges to the first. We will call this distance (by convention) $\Delta x$ (pronounced “delta $x$”). Thus instead of taking the point $(x, f(x))$ and $(b, f(b))$ and taking a limit as $x$ goes to $b$ (i.e. letting $x$ ’move towards $b$’ until they merge together), instead we will think of $x$ as being $b$ plus ’some distance’ which is represented by $\Delta x$... which we can write mathematically as $x = b + \Delta x$. So instead of $(x, f(x))$ we have $(b+\Delta x, f(b + \Delta x))$. So we can rewrite our limit as follows:

$$\lim \limits _{x\rightarrow b} \frac {f(x) - f(b)}{x-b} = \lim \limits _{\Delta x\rightarrow 0} \frac {f(b+\Delta x) - f(b)}{(\Delta x + b) - b} = \lim \limits _{\Delta x\rightarrow 0} \frac {f(b+\Delta x) - f(b)}{\Delta x}$$

Generally, most of the effort arises in trying to get rid of that $\Delta x$ in the bottom so that we can actually evaluate the limit. But rather than go through a number of clever tricks and techniques to simplify functions to accomplish this, we will instead take a new approach to our problem.

First we will develop rules to tackle core function types. This will allow us to know how to handle specific types of functions that we run into. Next we will develop techniques to separate combinations of functions into their individual parts. This will allow us to take functions that are sums, differences, products, or quotients of different core functions and break them apart, then apply our individual core functions from earlier to handle each part. Finally we will develop a rule to handle compositions of functions. Composition functions need a different technique than the algebraic combinations from the previous part, but once we can split apart compositions and algebraic combinations of functions into individual core function parts will allow us to break apart even very complicated functions into their most basic forms, and then compute derivatives of those basic components. Once we have all this, we will add a few more specialty techniques to our toolbag - all to avoid dealing with those monsterous difference quotients we saw at the start of this section.

Although the idea of the derivative is essentially the same as the instantaneous rate of change that we discussed in limits, we will be approaching the algebraic challenge involved in a very different way. This is the mathematical approach - we solve a more general problem, to make specific instances easier. This is good to keep in mind as we move forward... it may seem like we are making more trouble for ourselves by making the problem harder, but in reality we will end up with some relatively easy and straight forward results that allow us to bypass a whole lot of messy algebra and computation. A little more pain now, for a lot less pain later!