Relationships between a function, its first derivative, and second derivative is the entirety of Calc AB. You have to be super fluent in understanding their relationship. Among other things, this has been a struggle for my students over the years. Every year I try to improve the depth at which we approach this topic. Something I’ve liked is the introduction of tables when we start doing function analysis.
Previously, we had talked about the influence of the first derivative on the original function. Now we’ve added the second derivative to talk about curvature. Students are given a table of values for f’(x) and f”(x). We discuss what the values of these functions tell us about the unseen original. We use the trends in f’(x) to make a “rough draft” of our function, drawing line segments that increase or decrease as dictated by the table. With the information provided by f”(x), we can provide curvature to our rough draft. This helps set the stage for curve sketching when we start with the graph of the function later.