Related Rates — Real-World Chain Rule
Two or more quantities changing over time, connected by an equation.
Method:
1. Draw a picture and label variables
2. Write an equation connecting the variables
3. Differentiate both sides with respect to TIME (dt)
4. Plug in known values and solve for the unknown rate
Classic example: A ladder 10 ft long slides down a wall. The base moves out at 2 ft/sec. How fast is the top sliding down when the base is 6 ft from the wall?
x² + y² = 100. Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0.
When x = 6, y = 8. dx/dt = 2.
2(6)(2) + 2(8)(dy/dt) = 0. dy/dt = -24/16 = -1.5 ft/sec.
Common scenarios: Expanding circles, filling cones, moving shadows, approaching cars. Know the geometry formulas.
Reference:
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