The document discusses three key applications of derivatives in medical and biology: 1) Calculating the growth rate of tumors using an exponential growth model and taking the derivative of tumor volume with respect to time. The growth rate is directly proportional to the current tumor volume. 2) Calculating blood flow velocity and its gradient using the law of laminar flow. The velocity decreases from the center to the walls of blood vessels. 3) Detecting issues with blood pressure or blood vessel walls by analyzing the velocity gradient, with higher gradients potentially indicating constrictions.