When you talk graphs, the derivative function and its parent function can confuse you till your blue in the face. I myself have had the that happen to me before. You could be looking at the graph of the derivative function, looking for where the parent function is positive and accidentally take the slope from the graph of the derivative function, instead of looking for where on the graph it is above the x axis. An example of this is:
f (x) = x^2
f ' (x) = 2x
For this example the derivative function f ' (x) = 2x is a line. When you use it for applications on the parent function, remembering that this represents the slope of the parent function and that the y values are the slope not the slope of the derivative function. This is something that I remind myself when I attempt a problem involving the derivative function. I find that it helps to check your work after attempting the question because you never know if you made an "oops" or not.
this is my blog for the Derivative Unit