Application of Derivatives – JEE Mains Mathematics

• by Zenith LMS
• Updated April 2025

1. Introduction

Derivatives are powerful tools used to solve real-world problems involving rates of change, increasing/decreasing functions, and finding local maxima or minima of functions.

2. Rate of Change

• The derivative of a quantity with respect to time represents its rate of change.
• Used in physics, economics, and biology to calculate speed, growth, etc.
• Example:

  If s(t) is the position, then v(t) = ds/dt gives velocity.

3. Increasing and Decreasing Functions

• If f′(x) > 0 for an interval, then f(x) is increasing on that interval.

• If f′(x) < 0 for an interval, then f(x) is decreasing on that interval.

4. Tangents and Normals

• Equation of Tangent:

  y – y₁ = f′(x₁)(x – x₁)

• Equation of Normal:

  y – y₁ = –1/f′(x₁)(x – x₁)

5. Maxima and Minima

• Points where the function attains its maximum or minimum value.
• First Derivative Test:

  • If f′(x) changes from + to – at x = c, local maximum at x = c.

  • If f′(x) changes from – to + at x = c, local minimum at x = c.

• Second Derivative Test:

  • If f′(c) = 0 and f″(c) > 0, then local minimum at x = c.

  • If f′(c) = 0 and f″(c) < 0, then local maximum at x = c.

6. Optimization Problems

• Finding the maximum or minimum value of a quantity under given conditions.
• Examples:
  • Maximizing area, minimizing cost or surface area.
  • Real-world applications in design, economics, engineering.

7. Application in Economics and Science

• Marginal functions are defined as derivatives:

• Marginal Cost = dC/dx

  Marginal Revenue = dR/dx

  Marginal Profit = dP/dx

• Used in population growth models, chemical reaction rates, motion under gravity, etc.