Well, I haven't thought that I'll ask these kind of questions but well, really I do not really understand that.
Suppose we are making some secant lines in order to get the average rate of change and therefore we have points $x$ and $a$ – points of intersection with the function. Point $x$ is the point where we want to find an instantaneous rate of change. So what we are doing is making the second point closer and closer to $x$ and here is the problem.
We can do that from either left or right side. The second point can be a little smaller or bigger. We can take limit from the left or from the right ($+$/$-$). And as I know either way I will get the derivative. But actually the definition of the derivative is a TWO sided limit. So as we know from the limit properties in order to this to exist both the left and the right limits must exist and be equal. Why is the definition created like that even if I could find the derivative without bothering about all of these requirements? Do I need that constraint about the limits being equal? Why?
I am sure there are some reasonable arguments about why is that so even if I am not fully aware of them.
Best Answer
There are certainly examples where we could take the "derivative" from the left and right and get two different answers. For example, consider the absolute value function $f(x) = |x|$. As we approach $x=0$ from the right, all the secant lines have slope $1$; as we approach $x=0$ from the left, all the secant lines have slope $-1$.
In such an example, there is no well-defined tangent line (think about the graph of $|x|$). So we're interested in defining the derivative only where there is a well-defined tangent line, which is when the limits from the left and right of the secant slopes will be equal.