NCERT Mathematics, Class 12, textbook pages 147–186 (Part I)

How to help your child with Class 12 Maths Chapter 6, Application of Derivatives, at home

Rates, monotonicity and maxima–minima, explained for parents: a 60-minute study session, a quick challenge, a 5-day plan and the mistakes to watch for.

maxarea of rectangle, perimeter 40

NCERT Class 12 Maths Chapter 6 Application of Derivatives: what it teaches

Your teen uses derivatives for rates of change, finds where functions increase or decrease, and solves maxima and minima problems like the largest area for a given fence or cheapest container.

Why it matters later. Optimisation is used in economics, engineering and Class 12 linear programming.

See it: this chapter in pictures

V = 4/3 πr³dV/dt = 4πr² dr/dt= 400π at r = 10
Chain rule links rates.
f(x) = x³ − 3xf′(x) = 3x² − 3decreasing on (−1, 1)
Sign of f′ decides.
perimeter 40: A = x(20 − x)A′ = 20 − 2x = 0 → x = 10max area 100
Square gives the maximum.

You need

  • graph paper
  • string for fence problems
  • a calculator

Tonight’s 60 minutes

  1. A balloon’s radius grows at 1 cm/s. How fast is volume changing at r = 10?
  2. Find intervals where f(x) = x³ − 3x is increasing.
  3. Find the maximum area of a rectangle with perimeter 40 (a 10 × 10 square).
  4. Classify critical points with the second derivative.

Words to use

  • rate of change
  • increasing
  • decreasing
  • critical point
  • maximum
  • minimum
  • second derivative test

Watch for

  • Forgetting to check endpoints. On closed intervals compare endpoints too.
  • Not linking rates with the chain rule. dV/dt = dV/dr × dr/dt.

What it sounds like at home

A short example of the conversation. Your words will differ; the questions are what matter.

YouPerimeter 40: best rectangle?

Your childA 10 × 10 square: area 100.

A game for this week: Fence challenge

  1. Use a 40 cm string.
  2. Make rectangles and measure areas.
  3. Find the largest.
  4. Confirm with calculus.

A 5-day plan, 60 minutes a day

  1. Day 1Explain-back.
  2. Day 2Worked examples.
  3. Day 3Practice set.
  4. Day 4Real-life problem.
  5. Day 5Mixed quiz.

Everyday moments to practise

  • Minimising packaging cost.
  • Maximising profit.
  • Rates in physics.
  • Designing tanks.

Quick check

Ask these at the end of the week. Tap to see the answer.

Max area of a rectangle with perimeter 40?

100 (square 10 × 10).

Where is x³ − 3x decreasing?

−1 < x < 1.

dV/dt for a sphere at r = 10 if dr/dt = 1?

400π cm³/s.

Ready for the next chapter when your child…

  • Solves rate problems.
  • Finds monotonic intervals.
  • Solves maxima–minima problems.

Activities, wording and illustrations are Mathify Kids’ own, following the topics of NCERT’s Mathematics (Class 12). See the full Class 12 parent guide for every chapter.

Questions parents ask

How do I find maxima and minima?

Set f′(x) = 0, then use the second derivative or sign changes.

What are related rates?

Rates linked through a formula and the chain rule.

Are tangents and normals included?

Check the current syllabus; they may be reduced.