Derivative by the limit definition of the function f(x) = (-3x+7)/(-5x-7)

As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the
function at that point. If we differentiate a position function at a given time, we obtain the velocity at that time. It seems
reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about
the behavior of the function. However, the process of finding the derivative at even a handful of values using the techniques
of the preceding section would quickly become quite tedious.

The derivative function gives the derivative of a function at each point in the domain of the original function for which the
derivative is defined. We can formally define a derivative function as follows.


Let f be a function. The derivative function, denoted by f ′, is the function whose domain consists of those values
of x such that the following limit exists: limit as h goes to 0 of the difference quotient (f(x+h)-f(x))/h


A function f (x) is said to be differentiable at a if f ′(a) exists. More generally, a function is said to be differentiable
on S if it is differentiable at every point in an open set S, and a differentiable function is one in which f ′(x) exists on
its domain.

We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative
function, we could graph it. Given both, we would expect to see a correspondence between the graphs of these two
functions, since f ′(x) gives the rate of change of a function f (x) (or slope of the tangent line to f (x)).

Now that we can graph a derivative, let’s examine the behavior of the graphs. First, we consider the relationship between
differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there;
however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous
at a point and fail to be differentiable at the point for one of several reasons.