I am confused how to graph derivatives. I attached an image with the graph of a function and then I tried to graph the derivative. The part that gets me confused is figuring out the slope of the derivative. I know this is an approximation, but sometimes I see the graph of a function having a positive slope, and then the derivative has a negative slope. My teacher says to look at it like taking a limit from one side. Example, if a function approaches an undefined slope (like a cusps or a corner), then take the limit from both sides, so if the function approaches the cusp positively from the left the derivative is approaching negatively from the left. I guess it's just this reversal of perception that's confusing me at the moment.
Can someone check out my attached graphs and let me know if the derivates are correct? The graph on the left is the function(x) and the one in dark black on the right is f'(x). Thanks.