Rate of change of quantities
dy/dx is the rate of change of y with respect to x. When x and y both change with time, use dy/dt = (dy/dx) × (dx/dt).
What it means
A derivative measures how fast one quantity changes when another changes. If y = f(x), then dy/dx is the rate of change of y with respect to x. Its value at x = x₀ gives the rate at that one instant. When x and y both depend on time t, the chain rule links their rates: dy/dt = (dy/dx) × (dx/dt). The unit of a rate is (unit of the quantity) per (unit of time), for example cm²/s for an area. A quantity that is decreasing has a negative rate, so write it with a minus sign.
How to solve a rate-of-change problem
Use this order every time. It is the method the examiner marks.
- 1Name the variables. Write each given rate with its sign and unit, e.g. dr/dt = 3 cm/s.
- 2Write the formula that links the quantities, e.g. A = πr² for a circle.
- 3If the formula has an extra variable, remove it with the given relation (similar triangles, a fixed ratio).
- 4★Differentiate both sides with respect to t.
- 5Only now put in the values at the given instant.Putting numbers in before step 4 turns a changing quantity into a constant.
- 6Solve for the rate you want. Write it with its unit and say whether the quantity increases or decreases.
★ key step
Formulas you will use
| Shape and quantity | Formula | Rate with respect to time t |
|---|---|---|
| Circle: area | A = πr² | dA/dt = 2πr × dr/dt |
| Circle: circumference | C = 2πr | dC/dt = 2π × dr/dt |
| Sphere: volume | V = (4/3)πr³ | dV/dt = 4πr² × dr/dt |
| Sphere: surface area | S = 4πr² | dS/dt = 8πr × dr/dt |
| Cube of edge x: volume | V = x³ | dV/dt = 3x² × dx/dt |
| Cube of edge x: surface area | S = 6x² | dS/dt = 12x × dx/dt |
| Cone: volume | V = (1/3)πr²h | Write r in terms of h first, then differentiate |
Example 1: ripples in a temple tank
A stone falls into the still water of a temple tank. Circular ripples spread so that the radius grows at 3 cm/s. How fast is the area inside a ripple growing when its radius is 12 cm?
- Given: dr/dt = 3 cm/s. To find: dA/dt when r = 12 cm.
- Formula: A = πr²
- Differentiate with respect to t: dA/dt = 2πr × dr/dt
- Put r = 12 and dr/dt = 3: dA/dt = 2π × 12 × 3
- dA/dt = 72π cm²/s
The area grows at 72π cm²/s, about 226 cm²/s taking π ≈ 3.14.
Example 2: a sliding ladder
A ladder 13 m long leans against a wall. Its foot is pulled away from the wall along the ground at 6 cm/s. How fast is the top sliding down the wall when the foot is 5 m from the wall?
- Let x be the distance of the foot from the wall and y the height of the top, in m. Given dx/dt = 6 cm/s.
- By Pythagoras' theorem, x² + y² = 13² = 169.
- When x = 5, y² = 169 − 25 = 144, so y = 12.
- Differentiate with respect to t: 2x × dx/dt + 2y × dy/dt = 0
- So dy/dt = −(x/y) × dx/dt
- x/y has no unit, so dy/dt is in cm/s: dy/dt = −(5/12) × 6 = −2.5 cm/s
- The minus sign shows that the height is decreasing.
The top slides down the wall at 2.5 cm/s.
Example 3: marginal cost
The total cost, in rupees, of weaving x sarees in a week is C(x) = 0.01x³ − 0.3x² + 15x + 2000 (an example cost function). Find the marginal cost when 10 sarees are woven.
- Marginal cost MC = dC/dx
- dC/dx = 0.03x² − 0.6x + 15
- At x = 10: MC = 0.03 × 100 − 0.6 × 10 + 15
- MC = 3 − 6 + 15 = 12
The marginal cost is ₹ 12 per saree when 10 sarees are woven.
A doctor checks how fast a child's fever is falling each hour, and a small papad-making unit asks what one more packet adds to its cost. Both are rates of change.
1 marks: “The radius of a circle grows at 2 cm/s. At what rate does its circumference grow?” C = 2πr, so dC/dt = 2π × dr/dt = 2π × 2 = 4π cm/s. The rate is the same for every radius.
2 marks: “The total revenue, in rupees, from selling x units is R(x) = 3x² + 20x + 7 (an example). Find the marginal revenue when x = 8.” MR = dR/dx = 6x + 20. At x = 8, MR = 48 + 20 = ₹ 68 per unit.
3 marks: “The volume of a cube increases at 12 cm³/s. How fast is its surface area increasing when an edge is 8 cm long?” V = x³ gives dV/dt = 3x² × dx/dt, so 12 = 192 × dx/dt and dx/dt = 1/16 cm/s. S = 6x² gives dS/dt = 12x × dx/dt = 12 × 8 × (1/16) = 6 cm²/s.
Formula, differentiate, then substitute. Never put the numbers in before you differentiate.