The graph of a function's derivative can be constructed qualitatively from the graph of the function itself by using the correspondence between the derivative's output and the slope of the tangent line at each point: zeros of the derivative occur where the tangent line is horizontal, the derivative's sign matches the sign of the slope (positive where the function is increasing, negative where decreasing), and the derivative's magnitude reflects the steepness of the tangent line, reaching local extrema where the tangent line is steepest within a region. This belongs to differential calculus, within the study of the qualitative/geometric relationship between a function and its derivative, and connects to the broader theoretical fact that functions differing by an additive constant (vertical shifts) share an identical derivative.
The graph of a function's derivative can be constructed qualitatively from the graph of the function itself by using the correspondence between the derivative's output and the slope of the tangent li…