# Application of Derivatives, Class 12 Mathematics | SwaVid

Application of Derivatives, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers.

Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/application-of-derivatives

Source: https://www.swavid.com/cbse/class-12/maths/chapter/application-of-derivatives

# Application of Derivatives

## Formulas and definitions

## Key ideas

## Common mistakes

## Quick checks

## Keep going

## Questions parents ask

## Class 12 Mathematics on SwaVid

### Rate of change of quantities

### Increasing and decreasing functions

### Critical points

### First derivative test

### Second derivative test

### Absolute extrema on closed intervals

### Next chapter

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SwaVid › CBSE › Class 12 › Mathematics › Application of Derivatives

Notes written by SwaVid from the NCERT textbook.

Class 12 Mathematics, Chapter 6. Key ideas, formulas and common mistakes.

CBSE does not print marks for a single chapter.

Short answer. This chapter explains how to use derivatives to solve practical problems involving rates of change, increasing and decreasing intervals, and maxima and minima of functions. Students learn to determine optimal values and understand the geometric behaviour of curves using first and second derivatives.

Source: NCERT Mathematics Part I (Reprint 2026-27) , Chapter 6, Application of Derivatives

If a quantity y varies with respect to x such that y = f(x), then dy/dx represents the rate of change of y with respect to x. When two quantities x and y vary with another parameter t, the rate of change is computed as dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.

A function f is strictly increasing on an open interval (a, b) if f'(x) > 0 for all x in (a, b). It is strictly decreasing on (a, b) if f'(x) 0, then c is a point of local minimum. If f''(c) = 0, the test fails and the first derivative test must be used.

To find absolute maximum and minimum values of a continuous function on a closed interval [a, b], find all critical points in (a, b), calculate the function values at these critical points and at the endpoints a and b, and pick the greatest and least values.

Try each question first. Then tap it to see the answer.

Area A = πr²

dA/dr = d(πr²)/dr = 2πr

At r = 5 cm, dA/dr = 2π × 5 = 10π

10π cm²/cm

f'(x) = d/dx (2x² - 8x + 5) = 4x - 8

Set f'(x) > 0 gives 4x - 8 > 0

4x > 8 implies x > 2

(2, ∞)

f'(x) = 6x² - 30x + 36 = 6(x² - 5x + 6) = 6(x - 2)(x - 3)

Critical points in (1, 5) are x = 2 and x = 3

f(1) = 2(1) - 15(1) + 36(1) + 1 = 24

f(2) = 2(8) - 15(4) + 36(2) + 1 = 29

f(3) = 2(27) - 15(9) + 36(3) + 1 = 28

f(5) = 2(125) - 15(25) + 36(5) + 1 = 56

Maximum value among {24, 29, 28, 56} is 56

56

Integrals .

Continuity and Differentiability .

Application of Integrals .

The 13 chapters of Class 12 Mathematics .

CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all.

No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed.

Integrals.

SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works.

- Ch 6 NCERT book
- 80 marks, theory paper
- 3 quick checks
- Confusing strictly increasing with non-decreasing conditions . Use f'(x) > 0 for strictly increasing on an interval. Use f'(x) ≥ 0 when the question asks for an increasing function that allows constant stretches.
- Concluding that a test fails and giving up when f''(c) = 0 . When the second derivative test gives f''(c) = 0, revert to the first derivative test to examine the sign change of f'(x) around c.
- Forgetting to check boundary values for absolute extrema . Always evaluate f(x) at the endpoints a and b of the closed interval [a, b], not just at the interior critical points.
- Mixing up the sign of rate of change for decreasing quantities . If a quantity is decreasing over time, assign a negative sign to its rate of change, such as dx/dt = -2.

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