We have seen how to create, or derive, a new function $f'(x)$ from a function $f(x)$, summarized in the paragraph containing equation 2.1.1. Now that we have the concept of limits, we can make this more precise.

We know that $f'$ carries important information about the original function $f$. In one example we saw that $f'(x)$ tells us how steep the graph of $f(x)$ is; in another we saw that $f'(x)$ tells us the velocity of an object if $f(x)$ tells us the position of the object at time $x$. As we said earlier, this same mathematical idea is useful whenever $f(x)$ represents some changing quantity and we want to know something about how it changes, or roughly, the "rate'' at which it changes. Most functions encountered in practice are built up from a small collection of "primitive'' functions in a few simple ways, for example, by adding or multiplying functions together to get new, more complicated functions. To make good use of the information provided by $f'(x)$ we need to be able to compute it for a variety of such functions.

We will begin to use different notations for the derivative of a function. While initially confusing, each is often useful so it is worth maintaining multiple versions of the same thing.

Consider again the function $\ds f(x)=\sqrt{625-x^2}$. We have computed the derivative $\ds f'(x)=-x/\sqrt{625-x^2}$, and have already noted that if we use the alternate notation $\ds y=\sqrt{625-x^2}$ then we might write $\ds y'=-x/\sqrt{625-x^2}$. Another notation is quite different, and in time it will become clear why it is often a useful one. Recall that to compute the the derivative of $f$ we computed $$ \lim_{\Delta x\to0} {\sqrt{625-(7+\Delta x)^2} - 24\over \Delta x}. $$ The denominator here measures a distance in the $x$ direction, sometimes called the "run'', and the numerator measures a distance in the $y$ direction, sometimes called the "rise,'' and "rise over run'' is the slope of a line. Recall that sometimes such a numerator is abbreviated $\Delta y$, exchanging brevity for a more detailed expression. So in general, a derivative is given by $$ y'=\lim_{\Delta x\to0} {\Delta y\over \Delta x}. $$ To recall the form of the limit, we sometimes say instead that $$ {dy\over dx}=\lim_{\Delta x\to0} {\Delta y\over \Delta x}. $$ In other words, $dy/dx$ is another notation for the derivative, and it reminds us that it is related to an actual slope between two points. This notation is called Leibniz notation, after Gottfried Leibniz, who developed the fundamentals of calculus independently, at about the same time that Isaac Newton did. Again, since we often use $f$ and $f(x)$ to mean the original function, we sometimes use $df/dx$ and $df(x)/dx$ to refer to the derivative. If the function $f(x)$ is written out in full we often write the last of these something like this $$f'(x)={d\over dx}\sqrt{625-x^2} $$ with the function written to the side, instead of trying to fit it into the numerator.

Note. If you happen to know some "derivative formulas'' from an earlier course, for the time being you should pretend that you do not know them. In examples like the ones above and the exercises below, you are required to know how to find the derivative formula starting from basic principles. We will later develop some formulas so that we do not always need to do such computations, but we will continue to need to know how to do the more involved computations.

Sometimes one encounters a point in the domain of a function $y=f(x)$ where there is no derivative, because there is no tangent line. In order for the notion of the tangent line at a point to make sense, the curve must be "smooth'' at that point. This means that if you imagine a particle traveling at some steady speed along the curve, then the particle does not experience an abrupt change of direction. There are two types of situations you should be aware of—corners and cusps—where there's a sudden change of direction and hence no derivative.