You are the AI. Nobody hands you a slope formula — you put a finger on the curve and feel which way it tilts. Drag the gold point: the tangent's tilt is the derivative, right there. And your own dragging draws a brand-new curve underneath — the slope-collector, f′.
That's the whole trick — no derivative rules anywhere on this page. The machine picks two points a hair apart (0.01 left, 0.01 right) and computes plain rise over run, the same slope you met with points and lines. The tangent, every gold dot, and the completed f′ curve all come from exactly this arithmetic.