Class 12 Mathematics Chapter 6 Applications of Derivatives – Revision Notes

Applications of Derivatives turns the derivative into a practical tool, using it to find tangents and normals, approximate small changes, and locate a curve’s maxima and minima.

Last Updated: September 23, 2026

Key Concepts

• Rate of change: dy/dx as instantaneous rate.
• Increasing: f'(x)≥0; decreasing: f'(x)≤0.
• Tangent: y−y₀=f'(x₀)(x−x₀); Normal slope=−1/f'(x₀).

Approximation & Extrema

• Δy≈f'(x)Δx (differentials).
• Critical points: f'(x)=0.
• Second derivative test: f”<0 ⇒ local max; f”>0 ⇒ local min.
• Absolute extrema on [a,b]: compare critical points and endpoints.

One-Line Summary

Derivatives model rates of change, define tangents/normals, and identify increasing/decreasing behaviour and local/absolute extrema via first and second derivative tests.

Quick visual: a worked diagram from the full Solutions page, for reference.

Frequently Asked Questions

How are derivatives used to find whether a function is increasing or decreasing?
If the derivative of a function is positive over an interval, the function is increasing there, and if the derivative is negative, the function is decreasing, since the derivative represents the instantaneous rate of change.

How does the second derivative test help identify maxima and minima?
At a point where the first derivative is zero, a negative second derivative indicates a local maximum, a positive second derivative indicates a local minimum, avoiding having to check the sign change of the first derivative manually.

Written by Satish

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