Application of Derivatives

1) Hook — A Fun Real-Life Example

Imagine you are driving on the Mumbai-Pune Expressway. You notice your speedometer showing different speeds at different moments. How do you know when you are accelerating or slowing down? The concept of derivatives helps us understand how quantities change — like speed changing with time. In mathematics, derivatives help us find rates of change, which is very useful in real life, such as optimizing profits, minimizing costs, or determining the steepness of a hill.

2) Core Concepts — Application of Derivatives

The derivative of a function at a point gives the rate of change or the slope of the tangent to the curve at that point.

• Increasing/Decreasing Functions: If f'(x) > 0, the function is increasing at x; if f'(x) < 0, it is decreasing.
• Maxima and Minima: Points where the function reaches highest or lowest values locally.
• Finding Maxima and Minima: Steps:
  1. Find f'(x) and solve f'(x) = 0 to get critical points.
  2. Use f''(x) or test values around critical points to classify as max or min.

Example 1: Find the local maxima and minima of f(x) = x² - 6x + 8.

Answer: Local minimum at (3, -1). No local maximum.

Example 2: A farmer wants to fence a rectangular field along a river. The river side needs no fencing. If the farmer has 100 m of fencing, what dimensions should the field have to maximize the area?

Let the length along the river be x meters and the width perpendicular to the river be y meters.

• Perimeter fencing used: x + 2y = 100
• Area to maximize: A = x × y

Express y in terms of x: y = (100 - x)/2

Area function: A(x) = x × (100 - x)/2 = 50x - (x²)/2

Find A'(x) and set to zero:

Answer: Length = 50 m, Width = 25 m for maximum area.

3) Key Formulas/Rules

4) Did You Know?

Derivatives are not just for math class! The Indian Space Research Organisation (ISRO) uses derivatives to calculate the changing velocity and trajectory of rockets, ensuring they reach the right orbit. Without derivatives, space missions like Mangalyaan would not be possible!

5) Exam Tips

• Always check the domain: Sometimes critical points lie outside the domain given in the problem.
• Don’t forget the second derivative test: It confirms whether the critical point is max or min.
• Practice word problems: Many questions involve maximizing area, profit, or minimizing cost.
• Common mistakes: Forgetting to set derivative equal to zero, or mixing up max and min based on second derivative sign.
• Board pattern: Expect 2–3 questions on application of derivatives, including finding maxima/minima and rate of change.