Cheat sheet · Class 12 · CBSE · Commerce Mathematics
Application of Derivatives cheat sheet
The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.
CBSE Class 12 · Mathematics
Cheat sheet
Chapter 6Application of Derivatives
f′ = 0 finds critical points; f″ < 0 ⇒ maximum, f″ > 0 ⇒ minimum.
1. Rate of Change
— e.g. how fast a circle's area grows as its radius grows.
2. Increasing / Decreasing
- f′(x) > 0 ⇒ increasing; f′(x) < 0 ⇒ decreasing.
- Find intervals using critical points.
3. Maxima & Minima
| Test | Rule |
|---|---|
| First derivative | f′ changes + → − : local max; − → + : local min |
| Second derivative | f′(c) = 0, f″(c) < 0 : max; f″(c) > 0 : min |
| Absolute (closed interval) | compare values at critical points and end points |
Rectangle of fixed perimeter with max area is a square.
4. Key Points
- Point of inflection: f″ changes sign.
- Express the quantity in one variable first.
- Check endpoints for absolute extrema.
- Units of rates: per unit time.
Remember: Rates | increasing/decreasing | max–min tests
Application of Derivatives: questions and notes
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