Class 12 Mathematics Chapter 6: Applications of Derivatives
CBSE 2026–27 | Detailed NCERT-aligned resource
Learn how derivatives describe rates of change, increasing/decreasing behaviour, and maxima/minima. CBSE includes rate of change, increasing/decreasing functions, first derivative test, and second derivative test.
1. Rate of Change
If y=f(x), then dy/dx is the instantaneous rate of change of y with respect to x. If x depends on time, dy/dt=(dy/dx)(dx/dt).
Worked Example: Expanding Circle
A=πr². Differentiating with respect to time: dA/dt=2πr(dr/dt). If r=5 cm and dr/dt=2 cm/s, then dA/dt=20π cm²/s.
2. Increasing and Decreasing Functions
Where f′(x)>0, f is increasing; where f′(x)<0, f is decreasing. Find critical numbers first, then make a sign chart for f′.
| Sign of f′ | Behaviour |
|---|---|
| Positive | Increasing |
| Negative | Decreasing |
3. Critical Points
A critical point occurs where f′(x)=0 or f′ is undefined, provided f is defined there. It is a candidate for an extremum, not automatically a maximum or minimum.
4. First Derivative Test
f′: − → 0 → + = local minimum
5. Second Derivative Test
If f′(c)=0 and f″(c)<0, there is a local maximum at c. If f″(c)>0, there is a local minimum. If f″(c)=0, the test is inconclusive.
6. Full Worked Example
For f(x)=x³−3x²−9x+5, f′(x)=3(x−3)(x+1). Critical points are −1 and 3. The derivative is positive, negative, positive across the three intervals, so −1 is a local maximum and 3 is a local minimum.
7. Optimisation Example
A rectangle has perimeter 20 m. Let one side be x, so the other is 10−x. A=x(10−x)=10x−x². A′=10−2x=0 gives x=5. Since A″=−2, the maximum area is 25 m².
8. Exam Checklist
- Compare endpoints when finding absolute extrema on a closed interval.
- A root of f′ is not automatically an extremum.
- State units for physical rate problems.
- Use a sign chart whenever the first-derivative test is required.
9. Practice
- Find intervals of increase/decrease for x³−6x²+9x+2.
- Classify stationary points of x⁴−4x².
- Find a rate of change of volume of a sphere.
- Maximise the area of a rectangle with fixed perimeter.
Common Mistakes — Chapter 6
- Assuming f′(x)=0 automatically gives a maximum or minimum.
- Forgetting to compare endpoints for absolute extrema on a closed interval.
- Dropping units in rate-of-change problems.