TalentJR

Application of Derivatives – Class 12 Maths Chapter 6 Notes & NCERT PDF

Class 12MathematicsChapter 6NCERT book: Mathematics Part I - Class 12

Notes and a simple summary of Chapter 6 of the NCERT Class 12 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

🎧 Listen to the content

Chapter 6: Application of Derivatives

Introduction

Derivatives measure how fast one quantity changes with another. This chapter applies them to rates of change, to deciding where a function is increasing or decreasing, and to finding maxima and minima — the tools engineers, economists and scientists use to optimise. (In the rationalised syllabus, tangents/normals and approximations are not included; the chapter focuses on rate of change, monotonicity and extreme values.)

Key Concepts & Theorems

6.2 Rate of change of quantities

  • dy/dx (or f′(x₀)) is the rate of change of y with respect to x at x = x₀.
  • If both x and y depend on time t, dy/dt = (dy/dx)(dx/dt) — the chain rule links related rates.
  • Marginal cost = rate of change of total cost C(x) with respect to output x, i.e. C′(x); marginal revenue = R′(x).

    • 6.3 Increasing and decreasing functions

  • f is increasing on I if x₁ < x₂ ⇒ f(x₁) ≤ f(x₂); strictly increasing if f(x₁) < f(x₂). Similarly for decreasing.
  • Theorem 1: if f is continuous on [a, b] and differentiable on (a, b):

    • - f′(x) > 0 on (a, b) ⇒ f is increasing on [a, b];

    - f′(x) < 0 on (a, b) ⇒ f is decreasing;

    - f′(x) = 0 on (a, b) ⇒ f is constant.

  • Method: find f′(x), solve f′(x) = 0 to split the real line into intervals, then test the sign of f′ in each interval.

    • 6.4 Maxima and minima

  • Local maximum at c: f(c) ≥ f(x) for all x in some interval around c; local minimum similarly.
  • Theorem 2: at a local extremum inside an open interval, f′(c) = 0 or f is not differentiable at c. Such points are critical points.
  • Theorem 3 — First Derivative Test: at a critical point c,

    • - f′ changes from + to − ⇒ local maximum;

    - f′ changes from − to + ⇒ local minimum;

    - no change of sign ⇒ point of inflection (neither).

  • Theorem 4 — Second Derivative Test: if f′(c) = 0,

    • - f″(c) < 0 ⇒ local maximum;

    - f″(c) > 0 ⇒ local minimum;

    - f″(c) = 0 ⇒ test fails; use the first derivative test.

    6.4.1 Absolute maximum and minimum on a closed interval

  • Theorem 5: a continuous function on [a, b] always has an absolute maximum and an absolute minimum.
  • Method: find all critical points in the interval, evaluate f at those points and at the end points a and b, then pick the largest and smallest values.

    • Worked Examples

  • Rate of change (Example 1). A = πr² ⇒ dA/dr = 2πr = 10π cm²/cm at r = 5 cm.
  • Cube (Example 2). V = x³ with dV/dt = 9 cm³/s. At x = 10: dx/dt = 9/(3 × 100) = 0.03 cm/s. S = 6x² ⇒ dS/dt = 12x · dx/dt = 3.6 cm²/s.
  • Ripples (Example 3). dr/dt = 4 cm/s, r = 10 cm: dA/dt = 2πr · dr/dt = 80π cm²/s.
  • Rectangle (Example 4). dx/dt = −3, dy/dt = 2, x = 10, y = 6: perimeter changes at 2(−3 + 2) = −2 cm/min (decreasing); area changes at x·dy/dt + y·dx/dt = 20 − 18 = 2 cm²/min.
  • Marginal cost (Example 5). C(x) = 0.005x³ − 0.02x² + 30x + 5000 ⇒ C′(3) = 0.135 − 0.12 + 30 = ₹30.015 ≈ ₹30.02.
  • Monotonicity (Example 11). f(x) = 4x³ − 6x² − 72x + 30 ⇒ f′(x) = 12(x − 3)(x + 2). f is strictly increasing on (−∞, −2) and (3, ∞), strictly decreasing on (−2, 3).
  • Absolute extrema (Example 27). f(x) = 2x³ − 15x² + 36x + 1 on [1, 5]: critical points 2 and 3. f(1) = 24, f(2) = 29, f(3) = 28, f(5) = 56 ⇒ absolute maximum 56 at x = 5, absolute minimum 24 at x = 1.
  • Optimisation (Example 36). Squares of side x cut from a 3 m × 8 m sheet: V = x(3 − 2x)(8 − 2x). V′ = 0 gives x = 2/3 (x = 3 is impossible), and maximum volume = 200/27 m³.
  • Second derivative test. f(x) = x³ − 3x ⇒ f′(x) = 3x² − 3 = 0 at x = ±1; f″(x) = 6x. f″(−1) = −6 < 0 ⇒ local maximum 2 at x = −1; f″(1) = 6 > 0 ⇒ local minimum −2 at x = 1.

    • Why the derivative controls shape

    Think of f′(x) as the slope of the tangent at each point. While the slope stays positive the graph climbs; while it stays negative the graph falls. A peak or a valley can only occur where the slope becomes zero or the graph has a corner. The second derivative tells us how the slope itself is changing: if f″ is negative, the slope is decreasing, so the curve bends downward like a cap and a flat point is a peak. This single picture explains all the tests in the chapter.

    Formula / Key-Facts Sheet

  • Related rates: dy/dt = (dy/dx)(dx/dt)
  • Marginal cost = C′(x); marginal revenue = R′(x)
  • f′ > 0 ⇒ increasing; f′ < 0 ⇒ decreasing; f′ = 0 throughout ⇒ constant
  • Critical point: f′(c) = 0 or f′(c) does not exist
  • Second derivative test: f″(c) < 0 max, f″(c) > 0 min, f″(c) = 0 inconclusive
  • Closed interval: compare f at critical points and end points
  • Sum of two positive numbers fixed at 15 with minimum sum of squares ⇒ both equal 15/2

    • Board Exam Focus

  • 1–2 marks: rate of change (area of a circle, volume of a cube); marginal cost/revenue; show a simple function is increasing on R.
  • 3 marks: intervals of increase/decrease for a cubic; local maxima/minima by first or second derivative test.
  • 5 marks: optimisation word problems — open box from a sheet, cylinder in a sphere, fencing, minimum distance to a curve. Outline: (i) draw a figure and name variables, (ii) write the quantity to optimise as a function of one variable using the constraint, (iii) differentiate and solve f′ = 0, (iv) confirm max/min with f″ or sign change, (v) answer with units.

    • Key Terms

  • Rate of change (परिवर्तन की दर)
  • Marginal cost (सीमांत लागत)
  • Marginal revenue (सीमांत आय)
  • Increasing function (वर्धमान फलन)
  • Decreasing function (ह्रासमान फलन)
  • Critical point (क्रांतिक बिंदु)
  • Local maximum (स्थानीय उच्चतम)
  • Local minimum (स्थानीय निम्नतम)
  • Absolute maximum (निरपेक्ष उच्चतम)
  • Point of inflection (नति परिवर्तन बिंदु)
  • First derivative test (प्रथम अवकलज परीक्षण)
  • Second derivative test (द्वितीय अवकलज परीक्षण)
  • Optimisation (इष्टतमीकरण)

    • Common Mistakes

  • Forgetting the chain factor dx/dt in related-rate problems.
  • Giving a negative rate without saying the quantity is decreasing.
  • Missing the end points when finding absolute extrema on a closed interval.
  • Accepting a critical point outside the physical domain (like x = 3 in the box problem, which makes a side negative).
  • Concluding "no extremum" when f″(c) = 0 — the second derivative test only fails; use the first derivative test.
  • Writing intervals of increase with the wrong brackets or including points where f′ changes sign incorrectly.

    • 💡 Key Learning Points

    • ✓Interpret the derivative as a rate of change and solve related-rate problems using dy/dt = (dy/dx)(dx/dt).
    • ✓Compute marginal cost and marginal revenue as derivatives of cost and revenue functions.
    • ✓Find intervals where a function is increasing or decreasing from the sign of f′(x).
    • ✓Locate local maxima and minima using the first and second derivative tests.
    • ✓Find absolute extrema on a closed interval and solve real-life optimisation problems.

    👨‍🏫 Teaching Tips

    • →Open with a balloon being inflated: ask whether the radius grows at a constant rate if air enters at a constant rate.
    • →Use a sign chart of f′(x) on a number line for every monotonicity and first-derivative-test question.
    • →Build the open-box problem physically with A4 paper and compare measured volumes with the calculus answer.
    • →Stress the five-step optimisation template and make students write it before any calculation.
    • →Contrast local and absolute extrema with a graph that has a local maximum lower than an end-point value.
    • →Relate marginal cost to the cost of producing one more unit using a small numerical table.
    • →Give students a mixed set where they must decide between the first and second derivative tests.

    📋 Assessment Questions

    1. Can the student set up and solve a related-rates problem with correct units?
    2. Can the student find intervals of increase and decrease for a cubic function?
    3. Can the student apply the second derivative test and handle the case f″(c) = 0?
    4. Can the student find absolute maximum and minimum on a closed interval, including end points?
    5. Can the student model and solve an optimisation word problem?

    Ask the AI tutor about Class 12 Mathematics Chapter 6

    Get question answers, important questions, a quick MCQ quiz or a simple explanation of “Application of Derivatives”. Free — sign in with Google.

    How the AI tutor works →

    👩‍🏫 Class 12 Mathematics classes near you — online or at home

    Prefer a real teacher? TalentJR teachers take live online classes anywhere in India and visit homes in selected cities. Tell us what you need and our team will call you with timings and fees.

    • Live online classesSmall groups or 1:1 with a teacher, from home, any city.
    • Teacher at homeA TalentJR teacher visits your home (selected cities).
    • At a centreClasses at a TalentJR partner centre near you, where available.

    Keep learning

    All chapters in Mathematics Part I - Class 12

    1. 1. Relations and Functions
    2. 2. Inverse Trigonometric Functions
    3. 3. Matrices
    4. 4. Determinants
    5. 5. Continuity and Differentiability
    6. 6. Application of Derivatives