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CBSE · Class 12 · Mathematics

Rate of Change of Quantities (Ch 6: 6.2)

2 concepts · 16 practice questions with answers · Free to read

Rate of change with respect to a variable

From daily life

A speedometer shows how fast the distance is changing at this moment. In the same way dy/dx tells how fast y changes when x changes.

Simple definition

If y = f(x), then dy/dx is the rate of change of y with respect to x. It is positive if y increases with x and negative if y decreases. At a given point, substitute that value of x.

Formula: Rate of change of y with respect to x = dy/dx

Rule 1: Differentiate and substitute — Find the derivative first, then put in the value.

  • Easy: Area of a circle A = πr²: dA/dr = 2πr, which is 6π at r = 3.
  • Medium: Volume of a cube V = x³: dV/dx = 3x², which is 12 at x = 2.
  • Exam-type: Volume of a sphere V = (4/3)πr³: dV/dr = 4πr², which is 16π at r = 2.

Rule 2: Velocity as a rate — Position s(t) gives velocity s′(t).

  • Easy: s = t³ − 2t: velocity v = 3t² − 2, which is 10 at t = 2.
  • Medium: y = x² − 6x + 5 has rate zero where 2x − 6 = 0, i.e. at x = 3.
  • Exam-type: Circumference C = 2πr: dC/dr = 2π, a constant.
Remember

Substitute the numbers only after differentiating.

Common mistake

Putting r = 3 into A = πr² before differentiating.

Memory clue

First differentiate, then plug in.

Practice questions

Easy
  1. 1Find the rate of change of the area of a circle with respect to its radius at r = 3.
  2. 2Find the rate of change of the volume of a cube with respect to its edge when x = 2.
  3. 3Find dy/dx at x = 2 if y = 3x² + 2.
Medium
  1. 4A particle has position s = t³ − 2t. Find its velocity at t = 2.
  2. 5Find the rate of change of the area of a square with respect to its side at side 7.
  3. 6Find the rate of change of the volume of a sphere with respect to its radius at r = 2.
Challenging
  1. 7Find the rate of change of the circumference of a circle with respect to its radius.
  2. 8For y = x² − 6x + 5, find x where the rate of change is zero.
Show answers (try them yourself first)
  1. 12πr = 6π
  2. 23x² = 12
  3. 3dy/dx = 6x = 12
  4. 4v = 3t² − 2 = 10
  5. 5dA/dx = 2x = 14
  6. 64πr² = 16π
  7. 72π (a constant)
  8. 82x − 6 = 0, so x = 3

Rates of change with time (related rates)

From daily life

When a balloon is blown up, its radius and its volume both change with time. The rate of one can be found from the rate of the other.

Simple definition

If a quantity depends on another which depends on time, use the chain rule: dA/dt = (dA/dr) · (dr/dt). Write the formula, differentiate with respect to t, then substitute the given values.

Formula: dA/dt = (dA/dr) × (dr/dt)

Rule 1: Find the rate in time — Differentiate the formula with respect to t.

  • Easy: Circle radius grows at 2 cm/s: dA/dt = 2πr × 2, which is 20π cm²/s when r = 5.
  • Medium: Cube edge grows at 3 cm/s: dV/dt = 3x² × 3, which is 144 cm³/s when x = 4.
  • Exam-type: Balloon volume grows at 100 cm³/s: 4πr² dr/dt = 100, so at r = 5 the radius grows at dr/dt = 1/π cm/s.

Rule 2: Geometry link — Use a relation between the variables, then differentiate.

  • Easy: A square's side grows at 2 cm/s: its perimeter grows at 8 cm/s and its area at 2x × 2 = 40 cm²/s when x = 10.
  • Medium: A ladder: x² + y² = 25. Differentiating: 2x x′ + 2y y′ = 0.
  • Exam-type: If x = 3, y = 4 and x′ = 1 m/s, then y′ = −3/4 m/s (the top slides down).
Remember

Write a relation first, then differentiate with respect to t, and only then substitute.

Common mistake

Substituting the numbers before differentiating.

Memory clue

Relate, differentiate in t, substitute.

Practice questions

Easy
  1. 1The radius of a circle grows at 3 cm/s. How fast is the area growing when r = 4?
  2. 2The edge of a cube grows at 2 cm/s. How fast is the surface area growing when the edge is 5 cm?
  3. 3The side of a square grows at 4 cm/s. How fast is its area growing when the side is 6 cm?
Medium
  1. 4The radius of a sphere grows at 0.5 cm/s. How fast is the volume growing at r = 4 cm?
  2. 5A balloon is filled at 200 cm³/s. How fast does its radius grow when r = 10 cm?
  3. 6A point moves on y = x² so that x grows at 3 units/s. How fast does y grow at x = 2?
Challenging
  1. 7The radius of a circular oil spill grows at 2 cm/s. How fast does its circumference grow?
  2. 8A 5 m ladder leans on a wall. Its foot slides away at 1 m/s. How fast does the top slide down when the foot is 3 m from the wall?
Show answers (try them yourself first)
  1. 1dA/dt = 2πr × 3 = 24π cm²/s
  2. 2S = 6x², dS/dt = 12x × 2 = 120 cm²/s
  3. 3dA/dt = 2x × 4 = 48 cm²/s
  4. 4dV/dt = 4πr² × 0.5 = 32π cm³/s
  5. 54πr² dr/dt = 200, so dr/dt = 200 / (400π) = 1 / (2π) cm/s
  6. 6dy/dt = 2x × 3 = 12 units/s
  7. 7dC/dt = 2π × 2 = 4π cm/s
  8. 8x² + y² = 25: with x = 3, y = 4 the top slides down at 3/4 m/s