- The derivative f'(x) represents the slope of the original function f(x) at any point x.
- Where f(x) has a horizontal tangent (slope = 0), f'(x) crosses the x-axis. This occurs at the local maximum and minimum points of f(x).
- Where f(x) is increasing (positive slope), f'(x) is positive (above the x-axis).
- Where f(x) is decreasing (negative slope), f'(x) is negative (below the x-axis).
- The steepest parts of f(x) correspond to the highest and lowest points (local maxima/minima) of f'(x).
- The inflection point of f(x) (where concavity changes) corresponds to a local maximum or minimum of f'(x).
- The graph of f'(x) should reflect these changes in slope, crossing the x-axis at the critical points of f(x) and having peaks and valleys corresponding to the steepest slopes of f(x).
Parent Tip: Review the logic above to help your child master the concept of derivatives from graphs worksheet.
