Rate of change: If y = f(x), then dy/dx gives the rate of change of y with respect to x. If x and y are both functions of time t, then dy/dt = f'(x) * dx/dt (chain rule for related rates).
Common related rates problems:
- Expanding circle: dA/dt = 2pir * dr/dt
- Filling cone: V = (1/3)pir^2*h, with r/h = constant
- Ladder sliding: x^2 + y^2 = L^2, differentiate with respect to t
- Shadow problems: similar triangles + chain rule
JEE approach:
- Draw a diagram
- Identify all variables and given rates
- Find an equation relating the variables
- Differentiate with respect to t
- Substitute known values and solve for unknown rate
Example: A balloon's radius increases at 2 cm/s. Find the rate of volume increase when r = 5 cm. V = (4/3)pir^3. dV/dt = 4pir^2 * dr/dt = 4pi252 = 200pi cm^3/s.