Here, it is assumed that the function of the original function is unknown, only its graph.

Given the graph of a function, some facts can be told about the derivative, recalling the fact that the derivative function represents the instantaneous rate of change of the original function at different points.

Firstly, it is obvious that the original function and its derivative would usually have the same domain, except when certain x values (or range) where the function is not differentiable.

It is also logical that the graph of derivative follows that of the original. For instance, if the original function is a polynomial, so will the derivative, thus its graph would look like one.

We can then look at specific points. If there is(are) stationary point(s) found, we know that f^{\prime}(x)=0 at these point(s). This is because the graph f(x) at these points are somewhat horizontal, thus it’s instantaneous rate of change is 0.