Chapter Notes: Application of Derivatives (Class 12 CBSE - NCERT)

🔹 What is a Derivative?

A derivative represents the rate of change of a function with respect to a variable. In practical life, derivatives are used to analyze various types of changes such as increasing/decreasing behavior, maximum/minimum values, and more.

Key Applications of Derivatives

1. Rate of Change of Quantities

• Used to find how one quantity changes with respect to another.

• Example: Speed is the rate of change of distance with time.

Formula:
If $y = f(x)$, then
Rate of change = $\frac{dy}{dx}$

2. Increasing and Decreasing Functions

• A function $f(x)$ is:

  • Increasing on an interval if $f'(x) > 0$

  • Decreasing on an interval if $f'(x) < 0$

Steps to check:

1. Find $f'(x)$

2. Solve $f'(x) > 0$ or $f'(x) < 0$

3. Tangents and Normals

• Tangent: A straight line that just touches the curve at a point.

• Normal: A line perpendicular to the tangent at the point of contact.

Formulas:

• Slope of tangent at $x = a$: $m = f'(a)$

• Equation of tangent:
  $y - f(a) = f'(a)(x - a)$

• Slope of normal: $-\frac{1}{f'(a)}$

• Equation of normal:
  $y - f(a) = -\frac{1}{f'(a)}(x - a)$

4. Maxima and Minima (Optimization)

These help find the maximum or minimum value of a function.

Conditions:

• Let $f'(x) = 0$ at $x = c$

• If $f''(c) > 0$, then $f(c)$ is a minimum

• If $f''(c) < 0$, then $f(c)$ is a maximum

First Derivative Test:

• If $f'(x)$ changes from positive to negative at $x = c$ → Maximum

• If $f'(x)$ changes from negative to positive at $x = c$ → Minimum

🎯 Applications in Real Life (Word Problems)

• Economics: To find cost minimization or profit maximization.

• Physics: To determine velocity, acceleration.

• Geometry: Shortest distance problems, area optimization.

• Engineering: To optimize design structures.

🧮 Important Tips for Solving Problems

• Always begin with finding the derivative.

• Check for critical points (where $f'(x) = 0$ or undefined).

• Use sign table or second derivative to identify maxima/minima.

• For word problems:

  • Write the function in one variable.

  • Identify the quantity to be optimized.

  • Apply derivative rules and solve.

🧠 Formulas at a Glance